Joint value distribution of L-functions on the critical line
aa r X i v : . [ m a t h . N T ] F e b JOINT VALUE DISTRIBUTION OF L -FUNCTIONSON THE CRITICAL LINE SH ¯OTA INOUE AND JUNXIAN LI
Abstract.
In this paper, we discuss the joint value distribution of L -functions in a suitable class.We obtain joint large deviations results in the central limit theorem for these L -functions and somemean value theorems, which give evidence that different L -functions are “statistically independent”. Introduction
The value distribution of L -functions plays an important role in analytic number theory. A cele-brated theorem of Selberg [39] states that log ζ ( + it ) is approximately Gaussian distributed, namely,lim T →∞ T meas t ∈ [ T, T ] : log | ζ ( + it ) | q log log T > V = Z ∞ V e − u du √ π (1.1)for any fixed V . Throughout this paper, meas( · ) stands for the one-dimensional Lebesgue measure.Selberg also stated that primitive L -functions in the Selberg class are “statistically independent”without precise description for the independence (see [40, p.7]). Later, Bombieri and Hejhal [2] indeedshowed that normalized values of L -functions satisfying certain assumptions behave like independentGaussian variables, precisely,lim T →∞ T meas t ∈ [ T, T ] : log | L j ( + it ) | q n j log log T > V j for j = 1 , . . . , r = r Y j =1 Z ∞ V j e − u du √ π (1.2)for any fixed V , . . . , V r ∈ R under certain conditions of { L j } rj =1 , where n j are certain positive integersdetermined by L j . It is natural to ask if (1.1) and (1.2) still hold for a larger range of V . In fact, ifestimates similar to (1.1) holds for a larger range of V , then one could expect the growth of momentsof ζ ( + it ) as Z TT | ζ ( + it ) | k dt ≍ k T (log T ) k . This idea has been used by Soundararajan [41], where he proved the upper bound for all k ≥ T ) ε . However, the asymptotic (1.1)should not hold for arbitrarily large V , as observed by Radziwi l l [35], where he proved that (1.1) holdsfor V = o ((log log T ) / − ε ) and conjectured that (1.1) holds for V = o ( √ log log T ).In this paper, we derivate large deviations results for joint value distribution of L -functions andgive some further evidence of the “statistical independence” of L -functions in a general class, includingthe Riemann zeta function, Dirichlet L -functions, and L -functions attached to holomorphic and Maaßcusp forms. We obtain the expected upper bound (1.2) for V = o (cid:0) (log log T ) / (cid:1) (see Therorem 2.1).We also obtain weaker bounds when V ≪ √ log log T (see Theorem 2.2 and Theorem 2.3). As anapplication, we have the following result. Mathematics Subject Classification.
Primary 11M41; Secondary 60F10.
Key words and phrases.
Value distribution of L -functions, Central limit theorems, Mean value theorems, Largedeviations. Theorem 1.1.
Let { χ j } rj =1 be distinct primitive Dirichlet characters (where the case χ i = 1 for some ≤ i ≤ r is allowed). Then there exists some positive constant B depending on { χ j } rj =1 such that forany fixed sufficiently small positive real number k , Z TT (cid:18) min ≤ j ≤ r | L ( + it, χ j ) | (cid:19) k dt ≪ T (log T ) k /r + Bk , (1.3) Z TT (cid:18) max ≤ j ≤ r | L ( + it, χ j ) | (cid:19) − k dt ≫ T (log T ) k /r − Bk , (1.4) T (log T ) k /r + Bk ≪ Z TT exp (cid:18) k min ≤ j ≤ r Im log L ( + it, χ j ) (cid:19) dt ≪ T (log T ) k /r + Bk , and T (log T ) k /r − Bk ≪ Z TT exp (cid:18) − k max ≤ j ≤ r Im log L ( + it, χ j ) (cid:19) dt ≪ T (log T ) k /r + Bk . The above implicit constants depend on { χ j } rj =1 . If we assume the Riemann Hypothesis for L ( s, χ j ) for all j , then we have for any k > and ε > , Z TT (cid:18) min ≤ j ≤ r | L ( + it, χ j ) | (cid:19) k dt ≪ T (log T ) k /r + ε , Z TT (cid:18) max ≤ j ≤ r | L ( + it, χ j ) | (cid:19) − k dt ≫ T (log T ) k /r − ε ,T (log T ) k /r − ε ≪ Z TT exp (cid:18) k min ≤ j ≤ r Im log L ( + it, χ j ) (cid:19) dt ≪ T (log T ) k /r + ε , (1.5) and T (log T ) k /r − ε ≪ Z TT exp (cid:18) − k max ≤ j ≤ r Im log L ( + it, χ j ) (cid:19) dt ≪ T (log T ) k /r + ε . (1.6) Here, the above implicit constants depend on { χ j } rj =1 , k , and ε .Remark . Theorem 1.1 is a special case for Theorem 2.4 and Theorem 2.5, which are applicable tomore general L -functions such as L -functions associated with GL (2) cusp forms. Remark . It is now known that for 0 ≤ k ≤ Z TT | ζ ( + it ) | k dt ≍ k T (log T ) k , by the works of Heap-Radziwi l l-Soundararajan [14] and Heap-Soundararajan [15], so the unconditionalresult (1.3) is nontrivial when k is sufficiently small and 0 < k < B − /
2, if one of the L -functionsis chosen to be the Riemann zeta-function and r ≥
2. In particular, when k is sufficiently small and0 < k < B − /
2, we obtain the relation Z TT (cid:0) min (cid:8) | ζ ( + it ) | , | L ( + it, χ ) | (cid:9)(cid:1) k dt = o Z TT | ζ ( + it ) | k dt ! as T → + ∞ . This relation also holds for any k > ζ ( s ) and L ( s, χ ) is true. Remark . The conditional upper bound for the 2 k -th moment for Im log ζ ( + it ) was recently provedby Najnudel [34] using a different argument, but the lower bound remained unproved. Estimates (1.5),(1.6) give the lower bound and also recover the original Najnudel’s result. Remark . Gonek [8] showed a lower bound of the negative moment of the Riemann zeta-functionunder the Riemann Hypothesis. On the other hand, there were no unconditional results for the lowerbound. Estimate (1.4) gives an unconditional result for the negative moment for sufficiently small k . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 3 Proof sketch and outline of the paper.
Our proof starts with approximate formulae for L -functions that combines the formula of Selberg [39] and the hybrid formula of Gonek-Hughes-Keating[9] developed for the study of ζ ( s ). These formulae essentially contain a Dirichlet polynomial involvingprimes and explicit expressions involving zeros of these L -functions. After examining the contributionfrom the zeros, the joint value distribution of L -functions can be reduce to the joint value distributionof Dirichlet polynomials. To make clear of the scope of our methods, we work within a generalclass of L -functions satisfying suitable assumptions. These assumptions can be verified for Dirichlet L -functions and L -functions associated with GL (2) cusp forms. Note that we do not assume theRamanujan conjecture, but Hypothesis H of Rudnick-Sarnak [37]. The weaker assumption makes ourproof different in several aspects (see Section 3 and 4).The rest of the paper is organized as follows: In Section 2 we give the definition of the class of L -functions and state our results for these L -functions and their corresponding Dirichlet polynomials.In Section 3 we give the approximate formulas for L -functions in our class. In Section 4 we give theproofs of results for Dirichlet polynomials. In Section 5 and Section 6 we give the proof for resultsfor L -functions without and with the Riemann Hypothesis respectively. In Section 7, we give someremarks on other applications of our method. acknowledgement The authors would like to thank Winston Heap for his comments and remarks on earlier versionof the paper. The authors also thank Kenta Endo and Masahiro Mine for their useful comments.The first author is supported by Grant-in-Aid for JSPS Research Fellow (Grant Number: 19J11223).The second author thanks the Max Planck Institute for the financial support when attending theconference on zeta functions in Centre International de Rencontres Math´ematiques, where the projectwas started. 2.
Definitions and Results
We consider the class S † consisting of L -functions F ( s ) that satisfy the following conditions (S1)–(S5).(S1) (Dirichlet series) F ( s ) has a Dirichlet series F ( s ) = P ∞ n =1 a F ( n ) n s , which converges absolutelyfor σ > m F ∈ Z ≥ such that ( s − m F F ( s ) is entire of finiteorder.(S3) (Functional equation) F ( s ) satisfies the following functional equationΦ F ( s ) = ω F Φ F (1 − s ) , where Φ F ( s ) = γ ( s ) F ( s ) and γ ( s ) = Q s Q kℓ =1 Γ( λ ℓ s + µ ℓ ) , with λ ℓ > Q >
0, Re µ ℓ ≥ − ,and | ω F | = 1. Here we use the notation Φ F ( s ) = Φ F ( s ). The quantity d F = 2 P kℓ =1 λ ℓ isoften called the degree of F .(S4) (Euler product) F ( s ) can be written as F ( s ) = Y p F p ( s ) , F p ( s ) = exp ∞ X ℓ =1 b F (cid:0) p ℓ (cid:1) p ℓs ! , where b F ( n ) = 0 unless n = p ℓ with ℓ ∈ Z ≥ , and b F ( n ) ≪ n ϑ F for some ϑ F < .(S5) (Hypothesis H) For any ℓ ≥ X p | b F ( p ℓ ) log p ℓ | p ℓ < + ∞ . S. INOUE AND J. LI
This class of L -functions is different from the Selberg class in that we do not assume the RamanujanConjecture, which says(S3’) F ( s ) has the same functional equation as in (S3) with the condition Re µ ℓ ≥ ≤ ℓ ≤ k .(S5’) For every ε >
0, the inequality a F ( n ) ≪ F n ε holds.The set of L -functions satisfying (S1), (S2), (S3’), (S4), and (S5’) is called the Selberg class S . Itis believed that automorphic L -functions of GL ( n ) are in the Selberg class S , but this hasn’t beenproved due to the lack of the Ramanujan Conjecture. The weaker conditions in S † allow us to includemore automorphic L -functions of GL ( n ) unconditionally. The properties (S1)-(S4) for automorphic L -functions of GL ( n ) as well as their Rankin-Selberg convolutions can be found in [37, Section 2].Hypothesis H have been proved for 1 ≤ n ≤ n = 1 is trivial and the case for n = 2follows from the work of Kim-Sarnak [22]. See Rudnick-Sarnak [37, Proposition 2.4] for n = 3 andKim [21] for n = 4).To study the joint value distribution of L -functions in S † , we need the following assumption A . Assumption ( A ) . An r -tuple of Dirichlet series F = ( F , . . . , F r ) and an r -tuple of the numbers θ = ( θ , . . . , θ r ) ∈ R r satisfy A if and only if F , θ satisfy the following properties.(A1) (Selberg Orthonormality Conjecture) For any F j , we have X p ≤ x | a F j ( p ) | p = n F j log log x + O F j (1) , x → ∞ . (2.1)for some constant n F j >
0. For any pair F i = F j , X p ≤ x a F i ( p ) a F j ( p ) p = O F i ,F j (1) , x → ∞ . (2.2)(A2) For every i , there is at most one j = i such that F i = F j and in this case | θ i − θ j | ≡ π/ π ).(A3) (Zero Density Estimate) For every F j , there exists a positive constant κ F j such that, uniformlyfor any T ≥ / ≤ σ ≤ N F j ( σ, T ) ≪ F j T − κ Fj ( σ − / log T, (2.3)where N F ( σ, T ) is the number of nontrivial zeros ρ F = β F + iγ F of F ∈ S † with β F ≥ σ and0 ≤ γ F ≤ T . Remark . The Selberg Orthonormality Conjecture has been proved for L -functions associated withcuspidal automorphic representations of GL ( n ) unconditionally for n ≤ Remark . It is natural to assume (A2). This allows us to consider the joint distribution of log | F ( s ) | and Im log F ( s ). It can be seen that Re e − iθ log F ( s ) and Re e − iθ log F ( s ) can not be independentwhen | θ − θ | 6≡ π (mod π ). Remark . We require a strong zero density estimate (the exponent of log T is 1) close to the line σ = 1 /
2. Results of the shape (2.3) have been obtained for the Riemann zeta-function by Selberg[39] and for Dirichlet L -functions by Fujii [6]. For GL (2) L -functions, (A3) has been established byLuo [28] for holomorphic cusp forms of the full modular group (see [7, Section 7] for other congruencesubgroups). The proof should also be applicable for Maaß forms (see the remark in [28, p 141], seealso a weaker estimate in [38]). If we assume the Riemann Hypothesis for F j , then (2.3) holds for any κ F j > r be a fixed positive integer. For V = ( V , . . . , V r ) ∈ R r , θ = ( θ , . . . , θ r ) ∈ R r , and F = ( F , . . . , F r ) ∈ ( S † ) r satisfying assumption A , OINT VALUE DISTRIBUTION OF L -FUNCTIONS 5 we define S ( T, V ; F , θ ) := t ∈ [ T, T ] : Re e − iθ j log F j ( + it ) q n Fj log log T > V j for j = 1 , . . . , r , where the constants n F j are defined in (2.1). Let α F := min { r, − ϑ F ϑ F } , where ϑ F = max ≤ j ≤ r ϑ F j as defined in (S4). Here α F = 2 r if ϑ F = 0. We denote k z k = max ≤ j ≤ r | z j | . Throughout this paper,we write log x for log log log x .2.1. Results for large deviations.
The following theorem extends the result of Bombieri and Hejhal[2], where we show (1.2) holds for a larger range of V . Theorem 2.1.
Let θ = ( θ , . . . , θ r ) ∈ R r and F = ( F , . . . , F r ) ∈ ( S † ) r satisfy assumption A .Let A ≥ be a fixed constant. For any large T and any V = ( V , . . . , V r ) ∈ R r with k V k ≤ A (log log T ) / , we have T meas( S ( T, V ; F , θ )) (2.4)= (cid:18) O F ,A (cid:18) ( k V k + (log T ) )( k V k + 1) √ log log T + Q rk =1 (1 + | V k | )(log log T ) α F + (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u du √ π . Moreover, if θ ∈ [ − π , π ] r and k V k ≤ A (log log T ) / we have T meas( S ( T, V ; F , θ )) (2.5) ≤ (cid:18) O F ,A (cid:18) ( k V k + (log T ) )( k V k + 1) √ log log T + Q rk =1 (1 + | V k | )(log log T ) α F + (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u du √ π , and if θ ∈ [ π , π ] r , k V k ≤ a (log log T ) / , and Q rj =1 (1 + | V j | ) ≤ a (log log T ) α F + with a = a ( F ) > small enough, we have T meas( S ( T, V ; F , θ )) (2.6) ≥ (cid:18) − O F (cid:18) ( k V k + (log T ) )( k V k + 1) √ log log T + Q rk =1 (1 + | V k | )(log log T ) α F + (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u du √ π . Remark . For r = 1, F = ζ , and θ = 0, the asymptotic for k V k ≪ (log log T ) in (2.4) wasobtained by Radziwi l l [35], and the bound for k V k ≪ (log log T ) / in (2.5) was obtained by Inoue[19].It is reasonable to conjecture that the asymptotic in (2.4) holds for k V k = o ( √ log log T ) as spec-ulated in [35] for ζ ( s ). If we are only concerned with upper and lower bounds, we could extend therange of k V k further, which yields applications to moments of L -functions.We have the following unconditional results which extends the range of k V k in Theorem 2.1. Theorem 2.2.
Let F = ( F , . . . , F r ) ∈ ( S † ) r and θ = ( θ , . . . , θ r ) ∈ [ − π , π ] r satisfy assumption A .Let T be large. There exists some positive constant a = a ( F ) such that if θ ∈ [ − π , π ] r , we have T meas( S ( T, V ; F , θ )) ≪ F r Y j =1
11 + V j + 1(log log T ) α F + exp (cid:18) − V + · · · + V r O F (cid:18) k V k √ log log T (cid:19)(cid:19) S. INOUE AND J. LI for any V = ( V , . . . , V r ) ∈ ( R ≥ ) r satisfying k V k ≤ a (1+ V / m )(log log T ) / with V m = min ≤ j ≤ r V j ,and if θ ∈ (cid:2) π , π (cid:3) r , we have T meas( S ( T, V ; F , θ )) ≫ F r Y j =1
11 + V j exp (cid:18) − V + · · · + V r O F (cid:18) k V k √ log log T (cid:19)(cid:19) for k V k ≤ a (1 + V / m )(log log T ) / with Q rj =1 (1 + V j ) ≤ a (log log T ) α F + . Substituting V = V q nF log log T , . . . , V q nFr log log T ! to Theorem 2.2, we obtain the following corol-lary. Corollary 2.1.
Let F = ( F , . . . , F r ) ∈ ( S † ) r and θ = ( θ , . . . , θ r ) ∈ [ − π , π ] r satisfy assumption A . Set h F = n − F + · · · + n − F r . There exists a small constant a = a ( F ) such that if θ ∈ (cid:2) − π , π (cid:3) r ,we have T meas (cid:26) t ∈ [ T, T ] : min ≤ j ≤ r Re e − iθ j log F j ( + it ) > V (cid:27) ≪ F (cid:18) V / √ log log T ) r + 1(log log T ) α F + (cid:19) exp (cid:18) − h F V log log T (cid:18) O F (cid:18) V log log T (cid:19)(cid:19)(cid:19) for any ≤ V ≤ a log log T , and if θ ∈ (cid:2) π , π (cid:3) r , we have T meas (cid:26) t ∈ [ T, T ] : min ≤ j ≤ r Re e − iθ j log F j ( + it ) > V (cid:27) (2.7) ≫ F V / √ log log T ) r exp (cid:18) − h F V log log T (cid:18) O F (cid:18) V log log T (cid:19)(cid:19)(cid:19) for any ≤ V ≤ a min { log log T , (log log T ) α F r + + r } .Remark . When r = 1, F = ζ , and θ = 0, Jutila [20], using bounds on moments of ζ ( s ), hasproved meas (cid:8) t ∈ [ T, T ] : log | ζ ( + it ) | > V (cid:9) ≪ T exp (cid:18) − V log log T (cid:18) O (cid:18) V log log T (cid:19)(cid:19)(cid:19) uniformly for 0 ≤ V ≤ log log T . More recently, Heap and Soundararajan[15, Corollary 2] obtainedmeas (cid:8) t ∈ [ T, T ] : log | ζ ( + it ) | > V (cid:9) ≪ T exp (cid:18) − V log log T + O (cid:18) V log T √ log log T (cid:19)(cid:19) for √ log log T log T ≤ V ≤ (2 − o (1)) log log T using sharp bounds on moments of ζ ( s ). Our Theorem2.2 slightly improves Jutila’s bound by a factor of √ log log TV when √ log log T ≤ V ≤ a log log T forsome small constant a , but is weaker than [15] when √ log log T log T ≤ V ≤ (2 − o (1)) log log T .If we assume the Riemann Hypothesis for the corresponding L -functions, we can extend the rangeof k V k in Theorem 2.2 even further. Theorem 2.3.
Let F = ( F , . . . , F r ) ∈ ( S † ) r and θ = ( θ , . . . , θ r ) ∈ R r satisfy assumption A andassume that the Riemann Hypothesis is true for F , . . . , F r . Let V = ( V , . . . , V r ) ∈ ( R ≥ ) r and denote V m = min ≤ j ≤ r V j . Then there exists a = a ( F ) such that the following holds. If θ ∈ [ − π , π ] r and k V k ≤ a V / m (log log T ) / (log T ) / , we have T meas( S ( T, V ; F , θ )) (2.8) ≪ F (cid:18) V · · · V r + 1(log log T ) α F + (cid:19) exp (cid:18) − V + · · · + V r O F (cid:18) k V k √ log log T log k V k (cid:19)(cid:19) . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 7 If θ ∈ (cid:2) π , π (cid:3) r , k V k ≤ a V / m (log log T ) / (log T ) / and Q rj =1 V j ≤ a (log log T ) α F + , we have T meas( S ( T, V ; F , θ )) (2.9) ≫ F V · · · V r exp (cid:18) − V + · · · + V r − O F (cid:18) k V k √ log log T log k V k (cid:19)(cid:19) . Moreover, there exist some positive constants a = a ( F ) such that for any V ∈ ( R ≥ ) r with k V k ≥√ log log T and θ ∈ (cid:2) − π , π (cid:3) r , T meas( S ( T, V ; F , θ )) ≪ F exp (cid:0) − a k V k (cid:1) + exp (cid:16) − a k V k p log log T log k V k (cid:17) . (2.10)With r = 1, Theorem 2.3 slightly improves the bound in [30, Proposition 4.1] in the range of thefollowing corollary. Corollary 2.2.
Let F ∈ S † , and assume the Riemann Hypothesis for F . Let A ≥ , θ ∈ (cid:2) − π , π (cid:3) .Then, for any real number V with √ log log T ≤ V ≤ A (log log T ) / (log T ) / , we have T meas (cid:8) t ∈ [ T, T ] : Re e − iθ log F ( + it ) | > V (cid:9) ≪ A,F √ log log TV exp (cid:18) − V n F log log T (cid:19) . as T → ∞ . Moments of L -functions. We apply the large deviation results to obtain bounds for momentsof L -functions without and with the Riemann Hypothesis of the corresponding L -functions. Thefollowing theorem is a consequence of Theorem 2.2 without the Riemann Hypothesis. Theorem 2.4.
Let F = ( F , . . . , F r ) ∈ ( S † ) r and θ = ( θ , . . . , θ r ) ∈ R r satisfy assumption A . Set h F = n − F + · · · + n − F r . Then there exist some positive constants a = a ( F ) and B = B ( F ) such thatfor any fixed < k ≤ a , Z TT exp (cid:18) k min ≤ j ≤ r Re e − θ j log F j ( + it ) (cid:19) dt ≪ F T (log T ) k /h F + Bk (2.11) when θ ∈ (cid:2) − π , π (cid:3) r , and if ϑ F ≤ r +1 , we have Z TT exp (cid:18) k min ≤ j ≤ r Re e − iθ j log F j ( + it ) (cid:19) dt ≫ F T (log T ) k /h F − Bk (2.12) when θ ∈ (cid:2) π , π (cid:3) r . Here, the above implicit constants depend only on F . In particular, if ϑ F ≤ r +1 ,it holds that, for any < k ≤ a , Z TT (cid:18) min ≤ j ≤ r | F j ( + it ) | (cid:19) k dt ≪ F T (log T ) k /h F + Bk Z TT (cid:18) max ≤ j ≤ r | F j ( + it ) | (cid:19) − k dt ≫ F T (log T ) k /h F − Bk T (log T ) k /h F − Bk ≪ F Z TT exp (cid:18) k min ≤ j ≤ r Im log F j ( + it ) (cid:19) dt ≪ F T (log T ) k /h F + Bk and T (log T ) k /h F − Bk ≪ F Z TT exp (cid:18) − k max ≤ j ≤ r Im log F j ( + it ) (cid:19) dt ≪ F T (log T ) k /h F + Bk . Remark . The constant a could be computed explicitly by keeping track of the constants in theproof. It depends on the size of a F i ( p ), the constants ( n F j , O F i (1) , O F i ,F j (1)) in condition (A1), thedegree d F j = 2 P kℓ =1 λ ℓ , the constant κ F j , and the implicit constant in (2.3). We also need a ≪ F r . S. INOUE AND J. LI
Remark . We could also allow k → T → ∞ . In fact, as shown in the proof, the dependencyof the implicit constant in k is of size k √ log log T k √ log log T ) r .The restriction on k in Theorem 2.4 is due to the fact that the range of k V k in Theorem 2.2 is onlya small multiple of log log T . If we assume the Riemann Hypothesis for the corresponding L -functions,we can apply Theorem 2.3 to establish moment results for all k > Theorem 2.5.
Let F = ( F , . . . , F r ) ∈ ( S † ) r and θ = ( θ , . . . , θ ) ∈ [ − π , π ] r satisfy assumption A , and assume that the Riemann Hypothesis is true for F , . . . , F r . Let T be large, and put ε ( T ) =(log T ) − . Then there exists some positive constant B = B ( F ) such that for any k > , if θ ∈ (cid:2) − π , π (cid:3) r , we have Z TT exp (cid:18) k min ≤ j ≤ r Re e − θ j log F j ( + it ) (cid:19) dt ≪ k, F T (log T ) k /h F + Bk ε ( T ) (2.13) and if θ ∈ (cid:2) π , π (cid:3) r and ϑ F < r +1 , we have Z TT exp (cid:18) k min ≤ j ≤ r Re e − θ j log F j ( + it ) (cid:19) dt ≫ k, F T + T (log T ) k /h F − Bk ε ( T ) (2.14) In particular, if ϑ F < r +1 , it holds that, for any k > , ε > , Z TT (cid:18) min ≤ j ≤ r | F j ( + it ) | (cid:19) k dt ≪ ε,k, F T (log T ) k /h F + ε , Z TT (cid:18) max ≤ j ≤ r | F j ( + it ) | (cid:19) − k dt ≫ ε,k, F T (log T ) k /h F − ε ,T (log T ) k /h F − ε ≪ ε,k, F Z TT exp (cid:18) k min ≤ j ≤ r Im log F j ( + it ) (cid:19) dt ≪ ε,k, F T (log T ) k /h F + ε , and T (log T ) k /h F − ε ≪ ε,k, F Z TT exp (cid:18) − k max ≤ j ≤ r Im log F j ( + it ) (cid:19) dt ≪ ε,k, F T (log T ) k /h F + ε . Remark . One would expect that for any k >
0, it holds that Z TT exp (cid:18) k min ≤ j ≤ r Re e − iθ j log F j ( + it ) (cid:19) dt ≍ k T (log T ) k /h F (log log T ) ( r − / . It remains an interesting question to see if the bounds in Theorem 2.5 can be made sharp usingtechniques from Harper [11] under the corresponding Riemann Hypothesis.
Remark . An L -function is called primitive if it cannot be factored into L -functions of smallerdegree. It is conjectured that Z T | F ( + it ) | k dt ∼ C ( F, k ) T (log T ) k for some constant C ( F, k ) as T → ∞ , see [4, 12]. It is expected that the values of distinct primitive L -functions are uncorrelated, which leads to the conjecture Z T | F ( + it ) | k · · · | F r ( + it ) | k r dt ∼ C ( F , k ) T (log T ) k + ··· + k r for some constant C ( F , k ) as T → ∞ if F i = F j for i = j . This has be established for the product oftwo Dirichlet L -functions for k = k = 1 (see [12, 32, 43] and for some degree two L -functions when k = 1 and r = 1 (see [10, 45, 46]). For higher degree L -functions and higher values of k , obtainingthe asymptotic formula seems to be beyond the scope of current techniques. An upper bound of thiskind has been established by Milinovich and Turnage-Butterbaugh [30] for automorphic L -functions OINT VALUE DISTRIBUTION OF L -FUNCTIONS 9 of GL ( n ) under the Riemann Hypothesis for these L -functions. Modifying our approach, we canrecover their results (see Section 7) and obtain some weaker unconditional results for sufficiently small k i ’s when n ≤
2. Our results give some further evidence that distinct primitive L -functions are “statistically independent”. Remark . Our method as well as the works of Bombieri-Hejhal [2] and Selberg [39] requires astrong zero density estimate for L -functions. Unfortunately, the estimate has not been proved yet formany L -functions. There have been other methods to prove Selberg’s central limit theorem withoutthe strong zero density estimate such as Laurinˇcikas [24] and Radziwi l l-Soundararajan [36]. However,their methods do not apply to the central limit theorems for Im log ζ ( s ). Nevertheless, Hsu-Wong[17] proved a joint central limit theorem (for fixed V j ) for Dirichlet L -functions and certain GL (2) L -functions twisted by Dirichlet characters by using the method in [36]. The fact that the coefficientsof these L -functions satisfy | χ ( n ) | ≤ L -functions.2.3. Results for Dirichlet polynomials.
To prove the above theorems, we consider the Dirichletpolynomials associated with F . We need some notation before stating our results. Let Λ F ( n ) be thevon Mangoldt function associated with F defined by Λ F ( n ) = b F ( n ) log n . Let x = ( x , . . . , x r ) ∈ R r , z = ( z , . . . , z r ) ∈ C r , and F = ( F , . . . , F r ) be an r -tuple of Dirichlet series, and let θ = ( θ , . . . , θ r ) ∈ R r . When F satisfies (S4), we define P F ( s, X ) := X p ℓ ≤ X b F ( p ℓ ) p ℓs = X ≤ n ≤ X Λ F ( n ) n s log n , (2.15) σ F ( X ) := vuut X p ≤ X ∞ X ℓ =1 | b F ( p ℓ ) | p ℓ , (2.16) τ j ,j ( X ) = τ j ,j ( X ; F , θ ) := 12 X p ≤ X ∞ X ℓ =1 Re e − iθ j b F j ( p ℓ ) e − iθ j b F j ( p ℓ ) p ℓ ,K F , θ ( p, z ) := X ≤ j ,j ≤ r z j z j ∞ X ℓ =1 Re e − iθ j b F j ( p ℓ ) e − iθ j b F j ( p ℓ ) p ℓ , (2.17)and S X ( T, V ; F , θ ) := ( t ∈ [ T, T ] : Re e − iθ j P F j ( + it, X ) σ F j ( X ) > V j for j = 1 , . . . , r ) . Let {X ( p ) } p ∈P be a sequence of independent random variables on a probability space (Ω , A , P ) withuniformly distributed on the unit circle in C , where P is the set of prime numbers. Denote P F ( σ, X , X ) := X p ≤ X ∞ X ℓ =1 b F ( p ℓ ) X ( p ) ℓ p ℓσ , (2.18) M p,σ ( z ) = M p,σ ( z ; F , θ ) := E exp r X j =1 z j Re e − iθ j ∞ X ℓ =1 b F j ( p ℓ ) X ( p ) ℓ p ℓσ , (2.19)where E [] is the expectation. Finally, when F satisfies (S4) and (S5), we defineΞ X ( x ) = Ξ X ( x ; F , θ ) := exp X ≤ j Proposition 2.1. Let F = ( F , . . . , F r ) be an r -tuple of Dirichlet series and θ ∈ R r satisfy (S4),(S5), (A1), and (A2). Let T , X be large numbers with X (log log X ) r +1) ≤ T . Then there exists somepositive constant a = a ( F ) such that for V = ( V , . . . , V r ) ∈ R r with | V j | ≤ a σ F j ( X ) , T meas( S X ( T, V ; F , θ ))= (cid:18) O F (cid:18) Q rk =1 (1 + | V k | )(log log X ) α F + + 1 + k V k log log X (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u / du √ π . We could improve the range V j in Propositions 2.1 with a weaker error term. Proposition 2.2. Let F = ( F , . . . , F r ) be an r -tuple of Dirichlet series and θ ∈ R r satisfy (S4),(S5), (A1), and (A2). Let T , X be large numbers satisfying X (log log X ) r +1) ≤ T . Then for any V = ( V , . . . , V r ) ∈ ( R ≥ ) r with k V k ≤ (log log X ) r , we have T meas( S X ( T, V ; F , θ )) = (1 + E ) × Ξ X (cid:16) V σ F ( X ) , . . . , V r σ Fr ( X ) (cid:17) r Y j =1 Z ∞ V j e − u / du √ π , (2.21) where E satisfies E ≪ F exp C (cid:18) k V k√ log log X (cid:19) − ϑ F − ϑ F (cid:26) Q rk =1 (1 + V k )(log log X ) α F + + 1 √ log log X (cid:27) . Remark . In contrast to Proposition 2.1, we allow V j to be of size C √ log log X for arbitrarily large C , which is important in the proof of Theorem 2.5. We can prove an estimate similar to (2.21) forlarger V j , where we need to change the value of X suitably in this case. However, our main purposeis to prove Theorems 2.1, 2.2, and 2.3, and the case of larger V j is not required in their proofs. Forthis reason, we give only the case k V k ≤ (log log X ) r for simplicity.Applying Proposition 2.2, we can prove the following theorem. Theorem 2.6. Let F = ( F , . . . , F r ) be an r -tuple of Dirichlet series, and θ ∈ R r satisfy (S4),(S5), (A1), and (A2). Denote ϑ ∗ F = min ≤ j ≤ r ϑ F j . Then for any fixed k > there exist positiveconstants c = c ( k, F ) , c = c ( k, F ) and X = X ( F , k ) such that for any large T and X ≤ X ≤ max { c (log T log log T ) ϑ ∗ F , c T ) log log T log T } , we have T Z TT exp (cid:18) k min ≤ j ≤ r Re e − iθ j P F j ( + it, X ) (cid:19) dt = exp (cid:0) k H F ( X ) (cid:1) Q rj =1 σ F j ( X ) (cid:0) √ πkH F ( X ) (cid:1) r − q H F ( X ) Ξ X (cid:18) kH F ( X ) σ F ( X ) , . . . , kH F ( X ) σ F r ( X ) (cid:19) (1 + E ) , where H F ( X ) = 2 (cid:16)P rj =1 σ F j ( X ) − (cid:17) − , and E ≪ F ,k X ) α F − r − + log X √ log log X . Remark . When r = 1, these type of results can be obtained via mean value theorems of Dirichletseries (see Gonek-Hughes-Keating [9, Theorem 2] for ζ ( s ) and Heap [12, Theorem 2] for Dedekindzeta-functions associated with Galois extension number fields over Q ). Our method has the advantageto handle more than one L -functions, though the error term is weaker. Remark . The exact dependency of c , c on F and k are determined explicitly in the proof. When ϑ ∗ F = 0, the upper bound becomes X ≤ c (log T log log T ) where c ≪ F k . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 11 The asymptotic behavior can be further evaluated under some additional assumptions. Corollary 2.3. We use the same notation and assumptions as in Theorem 2.6. We further assume ϑ F < r +1 and the strong Selberg Orthonormality Conjecture, that is, X n ≤ X a F i ( p ) a F j ( p ) p = δ F i ,F j n F j log log X + e i,j + o (1) , X → + ∞ (2.22) for some constant e i,j . Here, δ F i ,F j is the Kronecker delta function, that is, δ F i ,F j = 1 if F i = F j , and δ F i ,F j = 0 otherwise. Then, as X → + ∞ , T Z TT exp k min ≤ j ≤ r Re e − iθ j X ≤ n ≤ X Λ F j ( n ) n / it log n dt ∼ C ( F , k, θ ) (log X ) k /h F (log log X ) ( r − / (2.23) for some positive constant C ( F , k, θ ) .Remark . When r = 1, F = ζ , and θ = 0 or π Corollary 2.3 recovers a result of Heap [13,Proposition 1] for real moments of partial Euler product with X < ( k − ε )(log T log log T ) .3. Approximate formulas for L -functions Approximate formulas for L -functions. In this section, we give an approximate formula forlog F ( s ). Here, we choose the branch of log F ( σ + it ) as follows. If t is equal to neither imaginaryparts of zeros nor poles of F , then we choose the branch by the integral log F ( σ + it ) = R σ + it ∞ + it F ′ F ( z ) dz .If t = 0 is equal to an imaginary part of a zero or a pole of F , then we take log F ( σ + it ) =lim ε ↓ log F ( σ + i ( t − sgn( t ) ε )). Here, sgn is the signum function. If there exists a pole or a zeroof F such that the imaginary part of zero, then we take log F ( σ ) = lim ε ↓ log F ( σ − ε ). Next weintroduce some notation. Notation. Let H ≥ f : R → [0 , + ∞ ) is mass one and supportedon [0 , f is a C ([0 , f belongs to C d − ( R ) and is a C d ([0 , d ≥ 2. For such f ’s, we define the number D ( f ), and functions u f,H , v f,H by D ( f ) = max (cid:8) d ∈ Z ≥ ∪ { + ∞} : f is a C d ([0 , (cid:9) ,u f,H ( x ) = Hf ( H log( x/e )) /x , and v f,H ( y ) = Z ∞ y u f,H ( x ) dx. Further, the function U ( z ) is defined by U ( z ) = Z ∞ u f,H ( x ) E ( z log x ) dx for Im( z ) = 0. Here, E ( z ) = E ( x + iy ) is the exponential integral defined by E ( z ) := Z + ∞ + iyx + iy e − w w dw = Z ∞ z e − w w dw. When Im( z ) = 0, then U ( x ) = lim ε ↑ U ( x + iε ).Let X ≥ ρ F = β F + iγ F be a nontrivial zero of F with β F , γ F realnumbers. Here, nontrivial zeros refer to the zeros of Φ F ( s ) that do not come from the Gamma factors γ ( s ). We also define σ X,t ( F ) for F = 1 and w X ( y ) by σ X,t ( F ) = 12 + 2 max | t − γ F |≤ X | βF − / | log X (cid:26) β F − , X (cid:27) , (3.1) w X ( y ) = ≤ y ≤ X , (log( X /y )) − X /y )) X ) if X ≤ y ≤ X , (log( X /y )) X ) if X ≤ y ≤ X . (3.2) Remark . The number σ X,t ( F ) is well-defined as a finite value for every X ≥ t ∈ R , F ∈ S † \ { } .Actually, by the same method as Conrey-Ghosh [3, Theorem 2], we see that d F ≥ F = 1 for any F ∈ S † . We also find that there are infinitely many zeros of F ∈ S † \ { } in the region 0 ≤ σ ≤ F does not have zeros for σ > + X ≤ σ X,t ( F ) ≤ .Then we have the following theorem, which is a generalization of [19, Theorem 1] in the case when F is the Riemann zeta-function ζ ( s ). Theorem 3.1. Assume F ∈ S † . Let D ( f ) ≥ , and H , X be real parameters with H ≥ , X ≥ .Then, for any σ ≥ / , t ≥ , we have log F ( s ) = X ≤ n ≤ X /H Λ F ( n ) v f,H (cid:0) e log n/ log X (cid:1) n s log n + X | s − ρ F |≤ X log(( s − ρ F ) log X ) + R F ( s, X, H ) , where the error term R F ( s, X, H ) satisfies R F ( s, X, H ) ≪ f m F X − σ ) + X − σ t log X (3.3)+ H ( σ X,t ( F ) − / X σ X,t ( F ) − σ ) + X σ X,t ( F ) − σ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) n σ X,t ( F )+ it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + d F log t for | t | ≥ t ( F ) with t ( F ) a sufficiently large constant depending on F .Remark . Note that in the above theorem we choose the branch of log ( s − ρ F ) as follows. If t = γ F ,then − π < arg ( s − ρ F ) < π , and if t = γ F , then arg ( s − ρ F ) = lim ε ↑ arg ( σ − β F + iε ). Remark . Theorem 3.1 is a modification and generalization of the hybrid formula by Gonek, Hughes,and Keating [9] to apply the method of Selberg-Tsang [44], where a different formula [44, Lemma 5.4]was used. With our formula, we can find the sign of the contribution from zeros to close s by the form P | s − ρ F |≤ X log(( s − ρ F ) log X ). This fact plays an important role in the proof of the theorems inSection 2 in a fashion similar to the work of Soundararajan [41].Theorem 3.1 can be obtained in the same method as the proof of [19, Theorem 1], where we needthe following proposition instead of [19, Proposition 1]. Proposition 3.1. Let F ∈ S † . Let X ≥ , H ≥ be real parameters. Then, for any s ∈ C , we have log F ( s ) = X ≤ n ≤ X /H Λ F ( n ) v f,H ( e log n/ log X ) n s log n + m ∗ F ( U (( s − 1) log X ) + U ( s log X )) − X ρ F ρ F =0 , U (( s − ρ F ) log X ) − ∞ X n =0 k X j =1 U (( s + ( n + µ j ) /λ j ) log X ) , OINT VALUE DISTRIBUTION OF L -FUNCTIONS 13 where the number m ∗ F is the integer such that the function ( s − m ∗ F F ( s ) is entire and not equal tozero at s = 1 . Using Theorem 3.1, we obtain the following propositions. Proposition 3.2. Let F ∈ S † satisfying (2.1) and (A3). Let σ ≥ / , and T be large. Then, thereexist positive constants δ F , A = A ( F ) such that for any k ∈ Z ≥ , ≤ X ≤ Y := T /k , T Z TT (cid:12)(cid:12)(cid:12)(cid:12) log F ( σ + it ) − P F ( σ + it, X ) − X | σ + it − ρ F |≤ Y log(( σ + it − ρ F ) log Y ) (cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ A k k k T δ F (1 − σ ) + A k k k X X
Suppose the same situation as Proposition 3.2. Then, there exist positive constants δ F , A = A ( F ) such that for any k ∈ Z ≥ , ≤ X ≤ Y := T /k , T Z TT | log F ( σ + it ) − P F ( σ + it, X ) | k dt ≤ A k k k T δ F (1 − σ ) + A k k k X X
We give the proofs of Proposition 3.1 andTheorem 3.1, but the proofs are almost the same as the proofs of [19, Proposition 2] and [19, Theorem1]. Therefore, we give the sketches only. Lemma 3.1. Let F ∈ S † \ { } . For all s ∈ C neither equaling to a pole nor a zero of F , we have F ′ F ( σ + it ) = X ρ F ρ F =0 , (cid:18) s − ρ F + 1 ρ F (cid:19) + η F − m F (cid:18) s − s (cid:19) − log Q (3.4) − ∞ X n =0 k X j =1 λ j Γ ′ Γ ( λ j s + µ j ) , where η F is a complex number and satisfies Re( η F ) = − Re P ρ F (1 /ρ F ) . In particular, for | t | ≥ t ( F ) ,we have F ′ F ( σ + it ) = X ρ F ρ F =0 , (cid:18) s − ρ F + 1 ρ F (cid:19) + O ( d F log | t | ) . (3.5) Proof. We obtain equation (3.4) in the same way as the proof of [31, eq. (10.29)]. Moreover, byapplying the Stirling formula to equation (3.4), we can also obtain equation (3.5). (cid:3) Lemma 3.2. Let F ∈ S † . For | t | ≥ t ( F ) , ≤ H ≤ | t | , we have X | t − γ F |≤ H ≪ d F H log | t | , (3.6) X | t − γ F |≥ H t − γ F ) ≪ d F log | t | H . (3.7) Proof. Applying the Stirling formula to Lemma 3.1, for | t | ≥ t ( F ), we haveRe (cid:18) F ′ F ( H + it ) (cid:19) = X ρ H − β F ( H − β F ) + ( t − γ F ) + O ( d F log | t | ) . On the other hand, it holds that X ρ F H − β F ( H − β F ) + ( t − γ F ) ≫ X | t − γ F |≤ H H , X ρ F H − β F ( H − β F ) + ( t − γ F ) ≫ X | t − γ F |≥ H H ( t − γ F ) . Since b F ( n ) log n ≪ F n / from (S4), we find that | ( F ′ /F )( H + it ) | = (cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =2 b F ( n ) log n/n H + it (cid:12)(cid:12)(cid:12)(cid:12) ≪ F ζ ( H − / − ≪ − H . Hence we obtain (3.6) and (3.7) for H ≥ 2. In addition, we immediately obtain these inequality thecase 1 ≤ H ≤ H = 2. (cid:3) Lemma 3.3. Let F ∈ S † . For any T ≥ t ( F ) , there exists some t ∈ [ T, T + 1] such that, uniformlyfor / ≤ σ ≤ , F ′ F ( σ + it ) ≪ F (log T ) . Proof. Using Lemma 3.2, we obtain this lemma in the same argument as the proof of [31, Lemma12.2]. (cid:3) Proof of Proposition 3.1. By using Lemma 3.3, we obtain Proposition 3.1 in the same method as theproof of [19, Proposition 1]. (cid:3) Lemma 3.4. Let d be a nonnegative integer with d < D = D ( f ) + 1 . Let s = σ + it be a complexnumber. Set X ≥ be a real parameter. Then, for any z = a + ib with a ∈ R , b ∈ R \ { t } , we have U (( s − z ) log X ) ≪ f,d X (1+1 /H )( a − σ ) + X a − σ | t − b | log X min ≤ l ≤ d ((cid:18) H | t − b | log X (cid:19) l ) . Proof. This lemma can be proved in the same way as [19, Lemma 2]. (cid:3) Lemma 3.5. Let s = σ + it be a complex number. Set X ≥ be a real parameter. Then, for anycomplex number z = a + ib with | t − b | ≤ / log X , we have U (( s − z ) log X ) = − log(( s − z ) log X ) + O (1) if | s − z | ≤ / log X , O (cid:0) X (1+1 /H )( a − σ ) + X a − σ (cid:1) if | s − z | > / log X .Proof. This lemma can be proved in the same way as [19, Lemma 3]. (cid:3) Proof of Theorem 3.1. It follows from Proposition 3.1, Lemma 3.4, and Lemma 3.5 thatlog F ( s ) = X ≤ n ≤ X /H Λ F ( n ) v f,H (cid:0) e log n/ log X (cid:1) n s log n + X | s − ρ F |≤ X log(( s − ρ F ) log X ) + R F ( s, X, H ) , OINT VALUE DISTRIBUTION OF L -FUNCTIONS 15 where R F ( s, X, H ) ≪ f,d m F ( X − σ ) + X − σ ) t log X + X | t − γ F |≤ X ( X β F − σ ) + X β F − σ )+ 1log X X | t − γ F | > X X β F − σ ) + X β F − σ | t − γ F | min ≤ l ≤ d ((cid:18) H | t − γ F | log X (cid:19) l ) . Hence, it suffices to show estimate (3.3) on the range | t | ≥ t ( F ). Following the proof of [19, Propo-sition 2], we see that it suffices to check X ρ F σ X,t ( F ) − / σ X,t ( F ) − β F ) + ( t − γ F ) ≪ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) n σ X,t ( F )+ it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + d F log | t | , (3.8)which can be shown by the same proofs as in [39, eq. (4.4); eq. (4.7)] by using equation (3.5) insteadof [39, Lemma 11]. (cid:3) Proof of Propositions 3.2 and Proposition 3.3.Lemma 3.6. Let T ≥ , and let X ≥ . Let k be a positive integer such that X k ≤ T . Then, for anycomplex numbers a ( p ) , we have Z TT (cid:12)(cid:12)(cid:12)(cid:12) X p ≤ X a ( p ) p it (cid:12)(cid:12)(cid:12)(cid:12) k dt ≪ T k ! X p ≤ X | a ( p ) | k . Here, the above sums run over prime numbers.Proof. By [44, Lemma 3.3], we have Z TT (cid:12)(cid:12)(cid:12)(cid:12) X p ≤ X a ( p ) p it (cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ T k ! X p ≤ X | a ( p ) | k + O k ! X p ≤ X p | a ( p ) | k ≤ T k ! X p ≤ X | a ( p ) | k + O k ! X k X p ≤ X | a ( p ) | k . Hence, we obtain this lemma when X k ≤ T . (cid:3) The next lemma is an analogue and a generalization of [44, Lemma 5.2]. Lemma 3.7. Let F ∈ S † \ { } be an L -function satisfying (A3). Let T be large, and κ F be the positiveconstant in (2.3) . For k ∈ Z ≥ , ≤ X ≤ T / , ξ ≥ with Xξ ≤ T κ F / , we have Z TT (cid:18) σ X,t ( F ) − (cid:19) k ξ σ X,t ( F ) − / dt ≪ F T k ξ X (log X ) k + C k k !log X (log T ) k − ! , where C = C ( F ) is a positive constant.Proof. By definition (3.1) of σ X,t ( F ), we obtain Z TT (cid:18) σ X,t ( F ) − (cid:19) k ξ σ X,t ( F ) − / dt (3.9) ≤ T ξ X (cid:18) X (cid:19) k + 2 k log X X T − X | βF − / | log X ≤ γ F ≤ T + X | βF − | log X β F ≥ / (cid:18) β F − (cid:19) k ( X ξ ) β F − . From the equation (cid:18) β F − (cid:19) k ( X ξ ) β F − = Z β F n ( σ − / k log ( X ξ ) + k ( σ − / k − o ( X ξ ) σ − dσ, we find that X T − X | βF − / | log X ≤ γ F ≤ T + X | βF − | log X β F ≥ / (cid:18) β F − (cid:19) k ( X ξ ) β F − ≤ X ≤ γ F ≤ Tβ F ≥ / Z β F n ( σ − / k log ( X ξ ) + k ( σ − / k − o ( X ξ ) σ − dσ ≤ Z n ( σ − / k log ( X ξ ) + k ( σ − / k − o ( X ξ ) σ − X ≤ γ F ≤ Tβ F ≥ σ dσ = Z n ( σ − / k log ( X ξ ) + k ( σ − / k − o ( X ξ ) σ − N F ( σ, T ) dσ. By assumption (A3), we can use the estimate N F ( σ, T ) ≪ F T − κ F ( σ − / log T , and so, for Xξ ≤ T κ F / , the above most right hand side is ≪ F T log T Z n ( σ − / k log ( X ξ ) + k ( σ − / k − o (cid:18) X ξ T κ F (cid:19) σ − dσ ≪ T C k k !(log T ) k − for some C = C ( F ) > 0. Hence, by this estimate and inequality (3.9), we obtain this lemma. (cid:3) Lemma 3.8. Let F be a Dirichlet series satisfying (S4), (S5), and (2.1) . Suppose there exists somepositive constant A such that | w ( n ) | ≤ (log n ) A . Then for any c ∈ R , σ ≥ , k ∈ Z ≥ and any X ≥ Z ≥ with X k ≤ T , we have Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Z ≤ n ≤ X b F ( n ) w ( n )(log( Xn )) c n σ + it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ T C k k k (log X ) c X Z ≤ p ≤ X | b F ( p ) w ( p ) | p σ k + T C k k k Z k (1 − σ ) (log X ) A + c ) k , where C = C ( F ) is some positive constant.Proof. Set K = 2(1 / − ϑ F ) − . Using | w ( n ) | ≤ (log n ) A , we find that for any σ ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ℓ>K X Z ≤ p ℓ ≤ X b F ( p ℓ ) w ( p ℓ )(log( Xp ℓ )) c p ℓ ( σ + it ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ F (log X ) c X p>Z X ℓ>K (log( p ℓ )) A p ℓ ( σ − / p (1 / − θ F ) ℓ ≪ F (log X ) c Z / − σ X p p − ε ≪ F (log X ) c Z / − σ . (3.10) OINT VALUE DISTRIBUTION OF L -FUNCTIONS 17 Therefore, by Lemma 3.6, we have Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Z ≤ n ≤ X b F ( n ) w ( n )(log( Xn )) c n σ + it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ T C k k k X ≤ ℓ ≤ K X Z /ℓ ≤ p ≤ X /ℓ | b F ( p ℓ ) w ( p ℓ )(log( Xp ℓ )) c | p ℓσ k + T C k Z k (1 − σ ) (log X ) ck for some constant C = C ( F ) > 0. From assumption (S5), we see that, for 2 ≤ ℓ ≤ K , X Z /ℓ ≤ p ≤ X /ℓ | b F ( p ℓ ) w ( p ℓ )(log( Xp ℓ )) c | p ℓσ ≤ (log X ) A + c ) Z − σ X p | b F ( p ℓ ) | p ℓ ≪ F Z − σ (log X ) A + c ) , which completes the proof of this lemma. (cid:3) Lemma 3.9. Let F be a Dirichlet series satisfying (S4), (S5), and (2.1) . Then for any x ≥ and c > , σ F ( x ) = X n ≤ x | b F ( n ) | n + O F (1) = X p ≤ x | b F ( p ) | p + O F (1) = n F log log x + O F (1) , (3.11) X n ≥ x | b F ( n ) | n c ≪ x − c , x → ∞ , where σ F ( x ) is defined by (2.16) .Proof. Set K = 2(1 / − ϑ F ) − . Similar to (3.10), we have X ℓ ≥ K X p | b F ( p ℓ ) | p ℓ ≪ F X p X ℓ p ≪ . From assumption (S5), we also have X ≤ ℓ ≤ K X p | b F ( p ℓ ) | p ℓ ≪ F , and thus ∞ X ℓ =2 X p | b F ( p ℓ ) | p ℓ ≪ F . Therefore, by assumption (2.1)2 σ F ( x ) = X p ≤ x ∞ X ℓ =1 | b F ( p ℓ ) | p ℓ = X n ≤ x | b F ( n ) | n + O F (1)= X p ≤ x b F ( p ) p + O F (1) = n F log log x + O F (1) , which is the assertion of (3.11).From partial summation and (3.11), we also find that X n ≥ x | b F ( n ) | n c = Z ∞ x ξ c − d X x ≤ n ≤ ξ | b F ( n ) | n = c Z ∞ x n F log log ξ − n F log log x + O F (1) ξ c ≪ F x − c . (cid:3) Lemma 3.10. Let F ∈ S † be an L -function satisfying (2.1) and (A3). Let T be large. Put δ F =min { , κ F } with κ F the positive constant in (2.3) . For any k ∈ Z ≥ , X ≥ with X ≤ T δ F /k , we have Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) n σ X,t ( F )+ it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt ≪ T C k k k (log X ) k , where w X is the smoothing function defined by (3.2) , and C = C ( F ) is a positive constant.Proof. For brevity, we write σ X,t ( F ) as σ X,t . X n ≤ X Λ F ( n ) w X ( n ) n σ X,t + it = X n ≤ X Λ F ( n ) w X ( n ) n / it − X n ≤ X Λ F ( n ) w X ( n ) n / it (1 − n / − σ X,t )= X n ≤ X Λ F ( n ) w X ( n ) n / it − Z σ X,t / X n ≤ X Λ F ( n ) w X ( n ) log nn α ′ + it dα ′ , Note that for 1 / ≤ α ′ ≤ σ X,t , (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log nn α ′ + it (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X α ′ − / Z ∞ α ′ X / − α X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it dα (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X σ X,t − / Z ∞ / X / − α (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) dα. Thus 1 T Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) n σ X,t + it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt ≪ C k ( S + S ) , (3.12)where S = 1 T Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) n / it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt, S = 1 T Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( σ X,t − 12 ) X σ X,t − / Z ∞ / X / − α (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) dα (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt. Since Λ F ( n ) = b F ( n ) log n and b F ( p ) = a F ( p ), we have X n ≤ X | Λ F ( n ) | n = (log X ) X p ≤ X | a F ( p ) | p − Z X ξξ X p ≤ ξ | a F ( p ) | p dξ + O (1)= n F (log X ) log log X − n F Z X log ξ × log log ξξ dξ + O F (cid:0) (log X ) (cid:1) = n F (log X ) log log X − n F (log X ) log log X + n F X ) + O F (cid:0) (log X ) (cid:1) ≪ F (log X ) . (3.13)Combining Lemma 3.8 with (3.13), we obtain S ≪ C k k k X n ≤ X | b F ( n ) | log nn k ≪ C k k k (log X ) k . (3.14) OINT VALUE DISTRIBUTION OF L -FUNCTIONS 19 By the Cauchy-Schwarz inequality and Lemma 3.7, when δ F ≤ κ F / 20, it holds that S ≤ Z TT ( σ X,t − / k X k ( σ X,t − / dt ! / ×× Z TT Z ∞ / X / − α (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) dα k dt ≪ T C k (log X ) k Z TT Z ∞ / X / − α (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) dα k dt , (3.15)for some constant C = C ( F ) > 0. We apply H¨older’s inequality to the innermost integral to obtain Z ∞ / X − α (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) dα k ≤ Z ∞ / X − α dα ! k − × Z ∞ / X − α (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) k dα = 1(log X ) k − Z ∞ / X / − α (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) k dα. Therefore, by Lemma 3.8 and (3.13), we find that Z TT Z ∞ / X − α (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) dα k dt ≤ X ) k − Z ∞ / X / − α Z TT (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X b F ( n ) log ( Xn ) log nn α + it (cid:12)(cid:12)(cid:12)(cid:12) k dt dα ≤ T C k k k (log X ) k − Z ∞ / X / − α (log X ) X n ≤ X | Λ F ( n ) | (log n ) n α k + (log X ) k dα ≪ T C k k k (log X ) k (log X ) X n ≤ X | Λ F ( n ) | n k + (log X ) k ≪ T C k k k (log X ) k . (3.16)Combining (3.12), (3.14), (3.15) and (3.16), we complete the proof of Lemma 3.10. (cid:3) Lemma 3.11. Let F ∈ S † be an L -function satisfying (2.1) and (A3). Let σ ≥ / , T be large. Let κ F , δ F be the same constants as in Lemma 3.10. There exists a positive constant C = C ( F ) such thatfor any k ∈ Z ≥ , ≤ X ≤ T δ F /k , Z TT X | σ + it − ρ F |≤ X k dt ≤ C k T − (2 σ − δ F + δF log X (cid:18) log T log X (cid:19) k , (3.17) and Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | σ + it − ρ F |≤ X log(( σ + it − ρ F ) log X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ ( Ck ) k T − ( σ − / δ F + δF log X (cid:18) log T log X (cid:19) k + . (3.18) Proof. From (3.1), there are no zeros of F with | σ + it − ρ F | ≤ X when σ ≥ σ X,t ( F ). Put ξ := T δ F /k .Note that ξ ≥ ≤ X ≤ T δ F /k = ξ . From these facts, we have X | σ + it − ρ F |≤ X ≤ ξ σ X,t ( F ) − σ X | t − γ F |≤ X σ ≥ / 2. By definition (3.1) and the line symmetry of nontrivial zeros of F , we find that X | t − γ F |≤ X ≤ X | t − γ F |≤ X β F ≥ / ≪ X | t − γ F |≤ X β F ≥ / ( σ X,t ( F ) − / ( σ X,t ( F ) − β F ) + ( t − γ F ) . Applying estimate (3.8) to the above right hand side, we obtain X | t − γ F |≤ X ≪ ( σ X,t ( F ) − / (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) n σ X,t ( F )+ it (cid:12)(cid:12)(cid:12)(cid:12) + d F log T (3.19)for t ∈ [ T, T ]. Noting Xξ k ≤ T κ F / and using Lemmas 3.7, 3.10, we have Z TT ( σ X,t ( F ) − / k ξ k ( σ X,t ( F ) − σ ) (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) n σ X,t ( F )+ it (cid:12)(cid:12)(cid:12)(cid:12) + d F log T k dt ≤ C k ξ k (1 / − σ ) ( (log T ) k Z TT ( σ X,t ( F ) − / k ξ k ( σ X,t ( F ) − / dt + Z TT ( σ X,t ( F ) − / k ξ k ( σ X,t ( F ) − / dt ! / Z TT (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ X Λ F ( n ) w X ( n ) n σ X,t ( F )+ it (cid:12)(cid:12)(cid:12)(cid:12) k dt / ) ≤ C k ξ k (1 / − σ ) T ξ k log X (cid:18) log T log X (cid:19) k + T ξ k log X k k/ ! ≤ C k T − (2 σ − δ F + δF log X (cid:18) log T log X (cid:19) k (3.20)for some constant C = C ( F ) > 0. Hence, we obtain estimate (3.17).Next, we show estimate (3.18). We find that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | σ + it − ρ F |≤ X log (( σ + it − ρ F ) log X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( g X ( s ) + π ) × X | σ + it − ρ F |≤ X , where g X ( s ) = g X ( σ + it ) is the function defined by g X ( s ) = log (cid:16) | ( σ + it − ρ s ) log X | (cid:17) if there exists a zero ρ F with | σ + it − ρ F | ≤ X ,0 otherwise. OINT VALUE DISTRIBUTION OF L -FUNCTIONS 21 Here, ρ s indicates the zero of F nearest from s = σ + it . By using the Cauchy-Schwarz inequality andestimate (3.17), we obtain Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | σ + it − ρ F |≤ X log(( σ + it − ρ F ) log X ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k dt (3.21) ≤ C k Z TT g X ( σ + it ) k dt + π k T ! / × T − ( σ − / δ F + δF log X (cid:18) log T log X (cid:19) k for some constant C = C ( F ) > 0. Moreover, we find that Z TT g X ( s ) k dt ≤ Z TT X | σ + it − ρ F |≤ X (cid:18) log (cid:18) | σ + it − ρ F | log X (cid:19)(cid:19) k dt ≤ X T − X ≤ γ F ≤ T + X Z γ F + X γ F − X (cid:18) log (cid:18) | t − γ F | log X (cid:19)(cid:19) k dt ≪ X X T − ≤ γ F ≤ T +1 Z (cid:18) log (cid:18) ℓ (cid:19)(cid:19) k dℓ ≪ F T log T log X Z (cid:18) log (cid:18) ℓ (cid:19)(cid:19) k dℓ. By induction, we can easily confirm that the last integral is equal to (2 k )!. Hence, we obtain Z TT g X ( s ) k dt ≪ F (2 k )! T log T log X . By substituting this estimate to inequality (3.21), we obtain this lemma. (cid:3) Proof of Proposition 3.2. Let f be a fixed function satisfy the condition of this paper (see Notation)and D ( f ) ≥ 2. Let σ ≥ / k ∈ Z ≥ , and 3 ≤ Z := T δ F /k , where δ F = min { κ F , } . It holds that Z σ Z,t ( F ) − σ ) + Z σ Z,t ( F ) − σ = Z / − σ ) · Z σ Z,t ( F ) − / + Z / − σ · Z σ Z,t ( F ) − / ≤ Z / − σ · Z σ Z,t ( F ) − / for σ ≥ / 2. Using this inequality and estimate (3.3) as H = 1, we find that there exists a positiveconstant C = C ( F ) such that (cid:12)(cid:12)(cid:12)(cid:12) log F ( σ + it ) − P F ( σ + it, Z ) − X | σ + it − ρ F |≤ Z log(( σ + it − ρ F ) log Z ) (cid:12)(cid:12)(cid:12)(cid:12) k (3.22) ≤ C k (cid:12)(cid:12)(cid:12)(cid:12) X Z 1, Λ F ( n ) = b F ( n ) log n , and a F ( p ) = b F ( p ), we can applyLemma 3.8 and Lemma 3.9 to bound the first sum in (3.22) so that1 T Z TT (cid:12)(cid:12)(cid:12)(cid:12) X Z 0. If Z < X ≤ Y , we see, using (2.1), that the sum on the right handside is ≪ F Z − σ = T δ F (1 − σ ) /k . If X < Z , it holds that the sum is ≤ P X
By Proposition 3.2, it suffices to show that there exists a positive constant A = A ( F ) such that Z TT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | σ + it − ρ F |≤ Y log(( σ + it − ρ F ) log Y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ≤ T − (2 σ − δ F A k k k (3.28)with Y = T /k . Put Z = T δ F /k . We write as X | σ + it − ρ F |≤ Y log(( σ + it − ρ F ) log Y )= X | σ + it − ρ F |≤ Z log(( σ + it − ρ F ) log Z ) − X Y < | σ + it − ρ F |≤ Z log(( σ + it − ρ F ) log Z )+ X | σ + it − ρ F |≤ Y log (cid:18) log Y log Z (cid:19) . It holds that − X Y < | σ + it − ρ F |≤ Z log(( σ + it − ρ F ) log Z ) + X | σ + it − ρ F |≤ Y log (cid:18) log Y log Z (cid:19) ≪ X | σ + it − ρ F |≤ Z . Hence, by using Lemma 3.11, we obtain (3.28). (cid:3) Proofs of results for Dirichlet polynomials In this section, we prove Propositions 2.1, 2.2, Theorem 2.6 and Corollary 2.3. Throughout thissection, we assume that F = ( F , . . . , F r ) is an r -tuple of Dirichlet series, and θ = ( θ , . . . , θ r ) ∈ R r .We also assume that {X ( p ) } p ∈P is a sequence of independent random variables on a probability space(Ω , A , P ) with uniformly distributed on the unit circle in C .4.1. Approximate formulas for moment generating functions. We will relate the momentgenerating function of ( P F j ( + it, X )) rj =1 to the moment generating function of random L -series.Recall that the Dirichlet polynomial P F j ( s, X ) is defined by (2.15). To do this, we work with a subsetof [ T, T ] such that the Dirichlet polynomials do not obtain large values. More precisely, define theset A = A ( T, X, F ) by A = r \ j =1 (cid:26) t ∈ [ T, T ] : | P F j ( + it, X ) | σ F j ( X ) ≤ (log log X ) r +1) (cid:27) . (4.1)We show that the measure of A is sufficiently close to T , and thus it enough to consider the momentgenerating function of ( P F j ( + it, X )) rj =1 on A .We first show that the measure of A is close to T . Lemma 4.1. Assume that F satisfies (S4), (S5), (A1). Let T, X be large with X (log log X ) r +1) ≤ T .Then there exists a positive constant b = b ( F ) such that T meas([ T, T ] \ A ) ≤ exp (cid:16) − b (log log X ) r +1) (cid:17) . Proof. By Lemmas 3.8, 3.9, there exists a constant C j depending only on F j such that Z TT | P F j ( + it, X ) | k dt ≤ T C kj k k X n ≤ X | b F j ( n ) | n k ≤ T (2 C j kσ F j ( X ) ) k (4.2)for 1 ≤ k ≤ − (log log X ) r +1) . This implies that1 T meas (cid:26) t ∈ [ T, T ] : | P F j ( + it, X ) | σ F j ( X ) > (log log X ) r +1) (cid:27) ≪ (cid:18) C j k (log log X ) r +1) (cid:19) k . Hence, there exists some constant C = C ( F ) > T meas([ T, T ] \ A ) ≤ (cid:18) Ck (log log X ) r +1) (cid:19) k . Choosing k = ⌊ (2 eC ) − (log log X ) r +1) ⌋ , we obtain this lemma. (cid:3) Next we consider the moment generating function of ( P F j ( + it, X )) rj =1 on A . Proposition 4.1. Assume that F , θ satisfy (S4), (S5), (A1), and (A2). Let T , X be large numberswith X (log log X ) r +1) ≤ T . For any z = ( z , . . . , z r ) ∈ C r with k z k ≤ X ) r , T Z A exp r X j =1 z j Re e − iθ j P F j ( + it, X ) dt (4.3)= Y p ≤ X M p, ( z ) + O (cid:16) exp (cid:16) − b (log log X ) r +1) (cid:17)(cid:17) , where M p, is the function defined by (2.19) . Here, b is a positive constant depending only on F . We prepare some lemmas for the proof of Proposition 4.1. Lemma 4.2. Let { b ( m ) } , . . . , { b n ( m ) } be complex sequences. For any q , . . . , q s distinct prime num-bers and any k , , . . . k ,n , . . . , k s, , . . . , k s,n ∈ Z ≥ , we have T Z TT s Y ℓ =1 (cid:16) Re b ( q k ,ℓ ℓ ) q − itk ,ℓ ℓ (cid:17) · · · (cid:16) Re b n ( q k n,ℓ ℓ ) q − itk n,ℓ ℓ (cid:17) dt = E " s Y ℓ =1 (cid:16) Re b ( q k ,ℓ ℓ ) X ( q ℓ ) k ,ℓ (cid:17) · · · (cid:16) Re b n ( q k n,ℓ ℓ ) X ( q ℓ ) k n,ℓ (cid:17) + O T s Y ℓ =1 n Y j =1 q k j,ℓ ℓ | b j ( q k j,ℓ ℓ ) | . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 25 Proof. Denote ψ h,ℓ = arg b h ( q k h,ℓ ℓ ). Then we can write (cid:16) Re b ( q k ,ℓ ℓ ) q − itk ,ℓ ℓ (cid:17) · · · (Re b n ( q k n,ℓ ℓ ) q − itk n,ℓ ℓ )= | b ( q k ,ℓ ℓ ) · · · b n ( q k n,ℓ ℓ ) | n X ε ,...,ε n ∈{− , } e i ( ε ψ ,ℓ + ··· + ε n ψ n,ℓ ) q − i ( ε k ,ℓ + ··· ε n k n,ℓ ) tℓ = | b ( q k ,ℓ ℓ ) · · · b n ( q k n,ℓ ℓ ) | n X ε ,...,ε n ∈{− , } ε k ,ℓ + ··· + ε n k n,ℓ =0 e i ( ε ψ ,ℓ + ··· + ε n ψ n,ℓ ) (4.4)+ | b ( q k ,ℓ ℓ ) · · · b n ( q k n,ℓ ℓ ) | n X ε ,...,ε n ∈{− , } ε k ,ℓ + ··· + ε n k n,ℓ =0 e i ( ε ψ ,ℓ + ··· + ε n ψ n,ℓ ) e − i ( ε k ,ℓ + ··· + ε n k n,ℓ ) t log q ℓ . Thus, we obtain s Y ℓ =1 (cid:16) Re b ( q k ,ℓ ℓ ) q − itk ,ℓ ℓ (cid:17) · · · (Re b n ( q k n,ℓ ℓ ) q − itk n,ℓ ℓ )= s Y ℓ =1 | b ( q k ,ℓ ℓ ) · · · b n ( q k n,ℓ ℓ ) | n X ε ,...,ε n ∈{− , } ε k + ··· + ε n k n =0 e i ( ε ψ ,ℓ + ··· + ε n ψ n,ℓ ) + E ( t ) , where E ( t ) is the sum whose the number of terms is less than 2 ns , and the form of each term is δ ′′ e it ( β log q + ··· + β s log q s ) with 0 ≤ | β ℓ | ≤ α ℓ := k ,ℓ + · · · k n,ℓ and β u = 0 for some 1 ≤ u ≤ s .Here, δ ′′ is a complex number independent of t , and satisfies | δ ′′ | ≤ − ns Q sℓ =1 Q nj =1 | b j ( q k j,ℓ ℓ ) | . Since | β log q + · · · + β s log q s | ≫ Q sℓ =1 Q nj =1 q − k j,ℓ ℓ , the integral of each term of E ( t ) is bounded by2 − ns Q sℓ =1 Q nj =1 q k j,ℓ ℓ | b j ( q k j,ℓ ℓ ) | . Hence, by this bound of E and the bound for the number of terms of E , we have Z TT E ( t ) dt ≪ s Y ℓ =1 n Y j =1 q k j,ℓ ℓ | b j ( q k j,ℓ ℓ ) | . Therefore, we have1 T Z TT s Y ℓ =1 (cid:16) Re b ( q k ,ℓ ℓ ) q − itk ,ℓ ℓ (cid:17) · · · (cid:16) Re b n ( q k n,ℓ ℓ ) q − itk n,ℓ ℓ (cid:17) dt = s Y ℓ =1 | b ( q k ,ℓ ℓ ) · · · b n ( q k n,ℓ ℓ ) | n X ε ,...,ε n ∈{− , } ε k + ··· + ε n k n =0 e i ( ε ψ ,ℓ + ··· + ε n ψ n,ℓ ) + O T s Y ℓ =1 n Y j =1 q k j,ℓ ℓ | b j ( q k j,ℓ ℓ ) | . On the other hand, by the independence of X ( p )’s, it follows that E " s Y ℓ =1 (cid:16) Re b ( q k ,ℓ ℓ ) X ( q ℓ ) k ,ℓ (cid:17) · · · (cid:16) Re b n ( q k n,ℓ ℓ ) X ( q ℓ ) k n,ℓ (cid:17) = s Y ℓ =1 E h(cid:16) Re b ( q k ,ℓ ℓ ) X ( q ℓ ) k ,ℓ (cid:17) · · · (cid:16) Re b n ( q k n,ℓ ℓ ) X ( q ℓ ) k n,ℓ (cid:17)i . As in (4.4), we can write (cid:16) Re b ( q k ,ℓ ℓ ) X ( q ℓ ) k ,ℓ (cid:17) · · · (cid:16) Re b n ( q k n,ℓ ℓ ) X ( q ℓ ) k n,ℓ (cid:17) = | b ( q k ,ℓ ℓ ) · · · b n ( q k n,ℓ ℓ ) | n X ε ,...,ε n ∈{− , } ε k ,ℓ + ··· + ε n k n,ℓ =0 e i ( ε ψ ,ℓ + ··· + ε n ψ n,ℓ ) + | b ( q k ,ℓ ℓ ) · · · b n ( q k n,ℓ ℓ ) | n X ε ,...,ε n ∈{− , } ε k ,ℓ + ··· + ε n k n,ℓ =0 e i ( ε ψ ,ℓ + ··· + ε n ψ n,ℓ ) X ( q ℓ ) − i ( ε k ,ℓ + ··· + ε n k n,ℓ ) . Since X ( p ) is uniformly distributed on the unit circle in C , we have E [ X ( p ) a ] = a = 0,0 otherwise (4.5)for any a ∈ Z . Hence, we obtain s Y ℓ =1 E h(cid:16) Re b ( q k ,ℓ ℓ ) X ( q ℓ ) k ,ℓ (cid:17) · · · (cid:16) Re b n ( q k n,ℓ ℓ ) X ( q ℓ ) k n,ℓ (cid:17)i = s Y ℓ =1 | b ( q k ,ℓ ℓ ) · · · b n ( q k n,ℓ ℓ ) | n X ε ,...,ε n ∈{− , } ε k ,ℓ + ··· + ε n k n,ℓ =0 e i ( ε ψ ,ℓ + ··· + ε n ψ n,ℓ ) , which completes the proof of the lemma. (cid:3) Lemma 4.3. Let { a ( p ) } p ∈P be a complex sequence. Then, for any k ∈ Z ≥ , X ≥ , we have E (cid:12)(cid:12)(cid:12)(cid:12) X p ≤ X a ( p ) X ( p ) (cid:12)(cid:12)(cid:12)(cid:12) k ≤ k ! X p ≤ X | a ( p ) | k . Proof. Since X ( p )’s are independent and uniformly distributed on the unit circle in C , it holds that E (cid:20) X ( p ) a · · · X ( p k ) a k X ( q ) b · · · X ( q ℓ ) b ℓ (cid:21) = ( p a · · · p a k k = q b · · · q b ℓ ℓ , p a · · · p a k k = q b · · · q b ℓ ℓ . It follows that E (cid:12)(cid:12)(cid:12)(cid:12) X p ≤ X a ( p ) X ( p ) (cid:12)(cid:12)(cid:12)(cid:12) k = X p ,...,p k ≤ Xq ,...,q k ≤ X a ( p ) · · · a ( p k ) a ( q ) · · · a ( q k ) E (cid:20) X ( p ) · · · X ( p k ) X ( q ) · · · X ( q k ) (cid:21) ≤ k ! X p ,...,p k ≤ X | a ( p ) | · · · | a ( p k ) | ≤ k ! X p ≤ X | a ( p ) | k , which completes the proof of the lemma. (cid:3) Lemma 4.4. Assume that F is a Dirichlet series satisfying (S4), (S5), and (2.1) . Let P F ( σ, X , X ) be the random Dirichlet polynomial defined by (2.18) . There exists a positive constant C = C ( F ) suchthat for any k ∈ Z ≥ , and any large X , we have E (cid:2) | P F ( , X , X ) | k (cid:3) ≪ ( Ckσ F ( X ) ) k . Proof. Using Lemma 4.3, we can prove this lemma in the same way as Lemma 3.8. (cid:3) OINT VALUE DISTRIBUTION OF L -FUNCTIONS 27 Proof of Proposition 4.1. Let T , X be large numbers such that X (log log X ) r +1) ≤ T . Let z =( z , . . . , z r ) ∈ C r with k z k ≤ X ) r . From (4.1), we have1 T Z A exp r X j =1 z j Re e − iθ j P F j ( + it, X ) dt = 1 T X ≤ k ≤ Y k ! Z A (cid:18) r X j =1 z j Re e − iθ j P F j ( + it, X ) (cid:19) k dt + O X k>Y k ! (cid:16) C (log log X ) r +5 / k z k (cid:17) k ! with Y = (log log X ) r +1) . Here, C = C ( F ) is some positive constant. We see that this O -term is ≪ exp (cid:0) − (log log X ) r +1) (cid:1) by the bound for k z k . Using the Cauchy-Schwarz inequality, we find that1 T Z A (cid:18) r X j =1 z j Re e − iθ j P F j ( + it, X ) (cid:19) k dt = 1 T Z TT (cid:18) r X j =1 z j Re e − iθ j P F j ( + it, X ) (cid:19) k dt + O T (meas([ T, T ] \ A )) / Z TT (cid:12)(cid:12)(cid:12)(cid:12) r X j =1 z j Re e − iθ j P F j ( + it, X ) (cid:12)(cid:12)(cid:12)(cid:12) k dt / . By Lemma 4.1, estimate (4.2), and the bound for k z k , this O -term is ≪ exp (cid:16) − (2 b ) − (log log X ) r +1) (cid:17) (cid:16) C k / (log log X ) r +1 / (cid:17) k for 0 ≤ k ≤ Y , where C = C ( F ) > T Z A exp r X j =1 z j Re e − iθ j P F j ( + it, X ) dt (4.6)= 1 T X ≤ k ≤ Y k ! Z TT (cid:18) r X j =1 z j Re e − iθ j P F j ( + it, X ) (cid:19) k dt + O exp (cid:16) − (2 b ) − (log log X ) r +1) (cid:17) X ≤ k ≤ Y ( C e (log log X ) r +1 / ) k k k/ . When X is sufficiently large, it follows that X ≤ k ≤ Y ( C e (log log X ) r +1 / ) k k k/ = X ≤ k ≤ (log log X ) r +2 ( C e (log log X ) r +1 / ) k k k/ + O (1) ≪ exp (cid:0) (log log X ) r +3 (cid:1) . Hence, the O -term on the right hand side of (4.6) is ≪ exp (cid:0) − (3 c ) − (log log X ) r +1) (cid:1) . Now, we can write Z TT (cid:18) r X j =1 z j Re e − iθ j P F j ( + it, X ) (cid:19) k dt = X ≤ j ,...,j k ≤ r z j · · · z j k X p ℓ ,...,p ℓkk ≤ X p ℓ / · · · p ℓ k / k × Z TT (cid:16) Re e − iθ j b F j ( p ℓ ) p − itℓ (cid:17) · · · (cid:16) Re e − iθ jk b F jk ( p ℓ k ) p − itℓ k k (cid:17) dt. From this equation and Lemma 4.2, we have1 T Z TT (cid:18) r X j =1 z j Re e − iθ j P F j ( + it, X ) (cid:19) k dt = E (cid:18) r X j =1 z j Re e − iθ j P F j ( , X , X ) (cid:19) k + O T X ≤ j ,...,j k ≤ r | z j · · · z j k | X p ℓ ,...,p ℓkk ≤ X | b F j ( p ℓ ) · · · b F jk ( p ℓ k k ) | p ℓ / · · · p ℓ k / k . Additionally, we see that this O -term is ≪ T r X j =1 | z j | X n ≤ X | b F j ( n ) | n / k ≤ ( CX ) k T ≤ C k T / ≤ exp (cid:16) − (3 b ) − (log log X ) r +1) (cid:17) for 0 ≤ k ≤ Y when T is sufficiently large. Therefore, we have1 T Z A exp r X j =1 z j Re e − iθ j P F j ( + it, X ) dt = X ≤ k ≤ Y k ! E r X j =1 z j Re e − iθ j P F j ( , X , X ) k + O (cid:16) exp (cid:16) − (3 c ) − (log log X ) r +1) (cid:17)(cid:17) = E exp r X j =1 z j Re e − iθ j P F j ( , X , X ) − X k>Y k ! E r X j =1 z j Re e − iθ j P F j ( , X , X ) k + O (cid:16) exp (cid:16) − (3 b ) − (log log X ) r +1) (cid:17)(cid:17) . The independence of X ( p )’s yields that E exp r X j =1 z j Re e − iθ j P F j ( , X , X ) = Y p ≤ X M p, ( z ) . Using Lemmas 3.9, 4.4, the Cauchy-Schwarz inequality, and the bound for k z k , we obtain E r X j =1 z j Re e − iθ j P F j ( , X , X ) k ≤ k k/ (cid:16) C (log log X ) r +1 / (cid:17) k OINT VALUE DISTRIBUTION OF L -FUNCTIONS 29 for some constant C = C ( F ) > 0. Therefore, it holds that X k>Y k ! E r X j =1 z j Re e − iθ j P F j ( , X , X ) k ≤ X k>Y (cid:0) C (log log X ) r +1 / (cid:1) k k k/ ≤ X k>Y (cid:18) C (log log X ) / (cid:19) k ≤ exp (cid:16) − (3 b ) − (log log X ) r +1) (cid:17) . Thus, we complete the proof of Proposition 4.1. (cid:3) Estimates for the main term of the moment generating function. We give some lemmasto estimate the main term in Proposition 4.1. Lemma 4.5. Assume that F , θ satisfy (S4), (S5). For any z = ( z , . . . , z r ) ∈ C r , σ ≥ / , we have M p,σ ( z ) = 1 + 14 X ≤ j ,j ≤ r z j z j ∞ X ℓ =1 Re e − iθ j b F j ( p ℓ ) e − iθ j b F j ( p ℓ ) p ℓσ + O ∞ X n =3 n ! r X j =1 | z j | ∞ X ℓ =1 | b F j ( p ℓ ) | p ℓσ n . Proof. We writeexp r X j =1 z j Re e − iθ j ∞ X ℓ =1 b F j ( p ℓ ) X ( p ) ℓ p ℓσ = 1 + ∞ X n =1 n ! X ≤ j ,...,j n ≤ r z j · · · z j n ∞ X ℓ ,...,ℓ n =1 Re e − iθ j b F j ( p ℓ ) X ( p ) ℓ p ℓ σ · · · Re e − iθ jn b F jn ( p ℓ n ) X ( p ) ℓ n p ℓ n σ . Let ψ h = arg b F h ( p ℓ h ) − θ j h . Then, using equation (4.5), we obtain E exp r X j =1 z j Re e − iθ j ∞ X ℓ =1 b F j ( p ℓ ) X ( p ) ℓ p ℓσ =1 + ∞ X n =1 n n ! X ≤ j ,...,j n ≤ r z j · · · z j n ∞ X ℓ ,...,ℓ n =1 | b F j ( p ℓ ) · · · b F jn ( p ℓ n ) | p ( ℓ + ··· + ℓ n ) σ X ε ,...,ε n ∈{− , } ε ℓ + ··· + ε n ℓ n =0 e i ( ε ψ + ··· + ε n ψ n ) . If n = 1, then the equation ε ℓ = 0 is always false, so the term corresponds to n = 1 vanishes. Theterm becomes corresponds to n = 2 is14 X ≤ j ,j ≤ r z j z j ∞ X ℓ =1 Re e − iθ j b F j ( p ℓ ) e − iθ j b F j ( p ℓ ) p ℓσ . The term corresponds to n ≥ ≤ n ! X ≤ j ,...,j n ≤ r | z j · · · z j n | ∞ X ℓ ,...,ℓ n =1 | b F j ( p ℓ ) · · · b F jn ( p ℓ n ) | p ( ℓ + ··· + ℓ n ) σ = 1 n ! r X j =1 | z j | ∞ X ℓ =1 | b F j ( p ℓ ) | p ℓσ n . This completes the proof of this lemma. (cid:3) Lemma 4.6. Assume that F satisfies (S4), (S5), (A1), and (A2). Put Ψ( z ) = Ψ( z ; F , θ ) := Y p M p, ( z )exp ( K F , θ ( p, z ) / . (4.7) Then, the infinity product is uniformly convergent on any compact set in C r . In particular, Ψ isanalytic on C r . Moreover, it holds that | Ψ( z ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) r Y j =1 exp − z j σ F j ( | z j | ) + O F (cid:18) | z j | + | z j | − ϑ F − ϑ F (cid:19)! (cid:12)(cid:12)(cid:12)(cid:12) (4.8) for any z = ( z , . . . , z r ) ∈ C , and that Ψ( x ) = r Y j =1 exp − x j σ F j ( | x j | ) + O F (cid:18) x j + | x j | − ϑ F − ϑ F (cid:19)! (4.9) for x ∈ R r . Furthermore, for any z = ( x + iu , . . . , x r + iu r ) ∈ C r satisfying x j , u j ∈ R with k u k ≤ ,we have Ψ( z ) = Ψ( x , . . . , x r ) r Y j =1 (cid:18) O F (cid:18) | u j | exp (cid:18) D k x k − ϑ F − ϑ F (cid:19)(cid:19)(cid:19) . (4.10) Here, D = D ( F ) is a positive constant.Proof. By the definition of K F , θ (see (2.17)), it holds that | K F , θ ( p, z ) | ≤ r k z k ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ . Here, b F ( p ℓ ) indicates the vector ( b F ( p ℓ ) , . . . , b F r ( p ℓ )). Using this estimate and Lemma 4.5, we findthat M p, ( z )exp( K F , θ ( p, z ) / ( K F , θ ( p, z ) + O ∞ X n =3 n ! r k z k ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ! n !) × ( − K F , θ ( p, z ) + O ∞ X n =3 n ! r k z k ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ ! n !) = 1 + O ∞ X n =3 ( r k z k ) n n ! ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ! n + ∞ X n =2 ( r k z k ) n n ! ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ ! n ! . (4.11)Let Y ≥ C ( k z k ) − ϑ F with C = C ( F ) > ϑ F , it holds that (cid:12)(cid:12)(cid:12)(cid:12) M p, ( z )exp( K F , θ ( p, z ) / − (cid:12)(cid:12)(cid:12)(cid:12) ≤ p ≥ Y . From this inequality and estimate (4.11), we obtain M p, ( z )exp( K F , θ ( p, z ) / 4) = exp log M p, ( z )exp( K F , θ ( p, z ) / ! = exp ( O ∞ X n =3 ( r k z k ) n n ! ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ! n + ∞ X n =2 ( r k z k ) n n ! ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ ! n !) (4.12)for any p ≥ Y . Set K = 2( − ϑ F ) − . Then, it holds that X ℓ>K k b F ( p ℓ ) k p ℓ/ ≪ F p − , and X ℓ>K k b F ( p ℓ ) k p ℓ ≪ F p − . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 31 Therefore, there exists some C = C ( F ) > ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ! n ≤ C n (cid:18) max ≤ ℓ ≤ K k b F ( p ℓ ) k n p ℓn/ + p − n (cid:19) ≤ C n X ≤ ℓ ≤ K k b F ( p ℓ ) k n p ℓn/ + p − n (4.13) ≤ C n r X j =1 X ≤ ℓ ≤ K k b F j ( p ℓ ) k n p ℓn/ + p − n and similarly that ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ ! n ≤ C n r X j =1 X ≤ ℓ ≤ K k b F j ( p ℓ ) k n p ℓn + p − n . (4.14)Using these inequalities and Lemma 3.9, we have for any Z ≥ , n ∈ Z X p>Z ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ! n ≤ C n r X j =1 X m>Z | b F j ( m ) | m / − ϑ F )( n − + C n Z − n (4.15) ≪ F C n Z − ( − ϑ F )( n − , n ≥ X p>Z ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ ! n ≤ C n r X j =1 X m>Z | b F j ( m ) | m − ϑ F )( n − + C n Z − n ≪ F C n Z − (1 − ϑ F )( n − , n ≥ . Applying these estimates to (4.12), we obtain Y p>Y M p, ( z )exp( K F , θ ( p, z ) / 4) = exp (cid:18) O F (cid:18) k z k Y − ϑ F (cid:19)(cid:19) = r Y j =1 exp (cid:18) O F (cid:18) | z j | Y − ϑ F (cid:19)(cid:19) (4.16)for any Y ≥ C ( k z k ) − ϑ F with C = C ( F ) a suitably large constant. In particular, we see that thisestimate implies the infinite product Q p M p, ( z )exp( K F , θ ( p, z ) / is uniformly convergent for z ∈ D with D anarbitrary compact set in C r . Hence, Ψ is analytic on C r .Next, we prove (4.8). Put L = C ( k z k ) − ϑ F . Then we divide the range of the product as Y p M p, ( z )exp ( K F , θ ( p, z ) / 4) = Y p ≤ L × Y p>L M p, ( z )exp ( K F , θ ( p, z ) / . If L < 2, then Q p ≤ L M p, ( z ) = 1 = exp (cid:18) O (cid:18) k z k − ϑ F − ϑ F (cid:19)(cid:19) . Next, we suppose L ≥ 2. Using Lemma4.5, we see that | M p, ( z ) | ≤ exp r X j =1 | z j | ∞ X ℓ =1 | b F j ( p ℓ ) | p ℓ/ ≤ exp r k z k ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ! . By (4.13) and the Cauchy-Schwarz inequality, we also find that X p ≤ L ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ≪ F max ≤ ℓ ≤ K X p ≤ L k b F ( p ℓ ) k p ℓ/ + 1 ≤ max ≤ ℓ ≤ K X p ≤ L k b F ( p ℓ ) k p ℓ / X p ≤ L / + 1 . Assumptions (S5), (A1), and the prime number theorem yield thatmax ≤ ℓ ≤ K X p ≤ L k b F ( p ℓ ) k p ℓ / X p ≤ L / ≪ F L / ≪ F k z k − ϑ F . Therefore, we obtain Y p ≤ L M p, ( z ) ≤ exp (cid:18) O F (cid:18) k z k − ϑ F − ϑ F (cid:19)(cid:19) = r Y j =1 exp (cid:18) O F (cid:18) | z j | − ϑ F − ϑ F (cid:19)(cid:19) for L ≥ 2. Combing this estimate with the estimate in the case L < 2, we have Y p ≤ L M p, ( z ) ≤ r Y j =1 exp (cid:18) O F (cid:18) | z j | − ϑ F − ϑ F (cid:19)(cid:19) (4.17)for all L ≥ 0. Also, it follows from the definitions of σ F ( X ), τ j ,j ( X ), and K F , θ ( p, z ) that X p ≤ L K F , θ ( p, z ) = 2 r X j =1 z j σ F j ( L ) + 4 X ≤ j 1. Hence, assumptions (S4), (S5), (A1), and (A2) yield that τ j ,j ( Z ) ≪ F Z ≥ j = j . By Lemma 3.9, σ F j ( Z ) = n F j Z + O F j (1) (4.20)for any Z ≥ 2. If | z j | ≤ k z k / , then z j σ F j ( L ) ≪ F k z k . If | z j | > k z k / , then we use (4.20) toobtain z j σ F j ( L ) = z j σ F j ( | z j | ) + z j X | z j |
4) = r Y j =1 exp − z j σ F j ( | z j | ) + O F ( | z j | ) ! . (4.21)Hence, we obtain (cid:12)(cid:12)(cid:12)(cid:12) Y p ≤ L M p, ( z )exp ( K F , θ ( p, z ) / (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) r Y j =1 exp − z j σ F j ( | z j | ) + O F (cid:18) | z j | + | z j | − ϑ F − ϑ F (cid:19)! (cid:12)(cid:12)(cid:12)(cid:12) . We also see that Y p>L M p, ( z )exp ( K F , θ ( p, z ) / 4) = exp (cid:0) O F (cid:0) k z k (cid:1)(cid:1) = r Y j =1 exp (cid:0) O F (cid:0) | z j | (cid:1)(cid:1) from (4.16). These estimates yield inequality (4.8). OINT VALUE DISTRIBUTION OF L -FUNCTIONS 33 Next, we show (4.9). It holds from the definition of M p, ( z ) thatexp − r X j =1 | x j | ∞ X k =1 | b F j ( p k ) | p k/ ≤ M p, ( x ) ≤ exp r X j =1 | x j | ∞ X k =1 | b F j ( p k ) | p k/ for any x = ( x , . . . , x r ) ∈ R r . Therefore, we have Y p ≤ L M p, ( x ) = exp O F k x k X p ≤ L ∞ X k =1 k b F ( p k ) k p k/ . Similarly to (4.17) and by this equation, we have Y p ≤ L M p, ( x ) = r Y j =1 exp (cid:18) O F (cid:18) | x j | − ϑ F − ϑ F (cid:19)(cid:19) . We can calculate the other parts similarly to the proof of (4.8), and obtain (4.9).Finally, we prove equation (4.10). Since Ψ is analytic on C r , we can writeΨ( x + iu , . . . , x r + iu r ) = ∞ X n =0 X k + ··· + k r = nk ,...,k r ≥ k ! · · · k r ! ∂ n Ψ( x , . . . , x r ) ∂z k · · · ∂z k r r ( iu ) k · · · ( iu r ) k r . It follows from estimates (4.8) and (4.9) that | Ψ( z , . . . , z r ) | ≤ Ψ( x , . . . , x r ) exp (cid:18) C (cid:18) k x k + k x k − ϑ F − ϑ F (cid:19)(cid:19) for some C = C ( F ) > | z − x | = · · · | z r − x r | = 2. Using this estimate and Cauchy’s integralformula, we find that ∂ n Ψ( x , . . . , x r ) ∂z k · · · ∂z k r r = k ! · · · k r !(2 πi ) r Z | z r − x r | =2 · · · Z | z − x | =2 Ψ( z , . . . , z r )( z − x ) k · · · ( z r − x r ) k r dz · · · dz r ≤ − ( k + ··· + k r ) k ! · · · k r !Ψ( x , . . . , x r ) exp (cid:18) C (cid:18) k x k + k x k − ϑ F − ϑ F (cid:19)(cid:19) . Hence, when k u k ≤ 1, we haveΨ( x + iu , . . . , x r + iu r ) = Ψ( x , . . . , x r ) (cid:18) O F (cid:18) k u k exp (cid:18) C k x k − ϑ F − θ F (cid:19)(cid:19)(cid:19) = Ψ( x , . . . , x r ) r Y j =1 (cid:18) O F (cid:18) | u j | exp (cid:18) C k x k − ϑ F − θ F (cid:19)(cid:19)(cid:19) , which completes the proof of (4.10). (cid:3) Lemma 4.7. Let Ξ X be the function defined by (2.20) . For x = ( x , . . . , x r ) ∈ ( R ≥ ) r , we have Ξ X ( x ) = r Y j =1 exp − x j σ F j ( | x j | ) + O F (cid:18) x j + | x j | − ϑ F − ϑ F (cid:19)! . (4.22) Proof. From (4.19), formula (4.9), and the definition of Ξ X , we haveΞ X ( x ) = r Y j =1 exp − x j σ F j ( | x j | ) + O F (cid:18) k x k + x j + | x j | − ϑ F − ϑ F (cid:19)! = r Y j =1 exp − x j σ F j ( | x j | ) + O F (cid:18) x j + | x j | − ϑ F − ϑ F (cid:19)! , which completes this lemma. (cid:3) Lemma 4.8. Assume that F satisfies (S4), (A1), and (A2). For z = ( z , . . . , z r ) ∈ C r , X ≥ C k z k − ϑ F + 3 with C = C ( F ) a sufficiently large positive constant, we have (cid:12)(cid:12)(cid:12)(cid:12) Y p ≤ X M p, ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) r Y j =1 exp z j (cid:0) σ F j ( X ) − σ F j ( | z j | (cid:1) + O F (cid:18) | z j | + | z j | − ϑ F − ϑ F (cid:19)! (cid:12)(cid:12)(cid:12)(cid:12) , (4.23) where σ F j ( X ) is defined by (2.16) . Moreover, there exists a positive constant b = b ( F ) such that,for any X ≥ and any z = ( z , . . . , z r ) ∈ C r with k z k ≤ b , we have Y p ≤ X M p, ( z ) = r Y j =1 (cid:0) O F (cid:0) | z j | (cid:1)(cid:1) exp z j σ F j ( X ) ! . (4.24) Furthermore, for any z = ( x + iu , . . . , x r + iu r ) ∈ C r with x j , u j ∈ R and k u k ≤ , and any X ≥ C k x k − ϑ F + 3 with C = C ( F ) a sufficiently large positive constant, we have Y p ≤ X M p, ( z ) (4.25)= Ξ X ( x ) r Y j =1 (cid:18) O F (cid:18) | u j | exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) + | z j | log X (cid:19)(cid:19) exp z j σ F j ( X ) ! , where Ξ X is the function defined by (2.20) , and D is the same constant as in Lemma 4.6.Proof. First, we prove (4.23). It holds that Y p ≤ X M p, ( z ) = Ψ( z ) Y p ≤ X exp ( K F , θ ( p, z ) / × Y p>X exp ( K F , θ ( p, z ) / M p, ( z ) . (4.26)Using (4.16), we have Y p>X exp ( K F , θ ( p, z ) / M p, ( z ) = r Y j =1 exp (cid:18) O F (cid:18) | z j | X − ϑ F (cid:19)(cid:19) (4.27)= r Y j =1 exp (cid:0) O F (cid:0) | z j | (cid:1)(cid:1) when X ≥ C k z k − ϑ F with C a suitably large constant. Also, as in the proof of (4.21), we find that Y p ≤ X exp ( K F , θ ( p, z ) / 4) = r Y j =1 exp z j σ F j ( X ) + O F (cid:0) | z j | (cid:1)! . Combing the above two estimates and inequality (4.8), we have estimate (4.23).Next, we prove (4.24). When k z k ≤ b with b = b ( F ) sufficiently small, it holds from Lemma 4.5that | M p, ( z ) − | ≤ , and that M p, ( z ) = 1 + 14 K F , θ ( p, z ) + O F k z k ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ! OINT VALUE DISTRIBUTION OF L -FUNCTIONS 35 with b F ( p ℓ ) = ( b F ( p ℓ ) , . . . , b F r ( p ℓ )). Using these and estimate (4.15), we obtain X p ≤ X log M p, ( z ) = X p ≤ X K F , θ ( p, z ) + O F k z k ∞ X ℓ =1 k b F ( p ℓ ) k p ℓ/ ! = 14 X p ≤ X K F , θ ( p, z ) + O F (cid:0) k z k (cid:1) . Similarly to (4.18), we also have14 X p ≤ X K F , θ ( p, z ) = r X j =1 z j σ F j ( X ) + O F (cid:0) k z k (cid:1) . Hence, it holds that X p ≤ X log M p, ( z ) = r X j =1 z j σ F j ( X ) + O F (cid:0) k z k (cid:1) = r X j =1 z j (cid:0) σ F j ( X ) + O F (1) (cid:1) , which completes the proof of (4.24).Finally, we prove (4.25). It follows from equations (4.26) and (4.27) that Y p ≤ X M p, ( z ) = Ψ( z ) exp X p ≤ X K F , θ ( p, z ) r Y j =1 (cid:18) O F (cid:18) | z j | X − ϑ F (cid:19)(cid:19) . for X ≥ C k x k − ϑ F + 3 with C = C ( F ) sufficiently large. Moreover, by equation (4.18), it holds thatexp X p ≤ X K F , θ ( p, z ) = exp r X j =1 z j σ F j ( X ) + X ≤ j In this section, we prove Propositions 2.1 and Proposition2.2. Proposition 2.2 is a generalization of Radziwi l l’s result [35, Proposition 2]. Radziwi l l adoptedTenenbaum’s formula [42] which relates the measure of the large values of the Dirichlet polynomialto a line integral. If we follow the same approach, we will need to assume Ramanujan’s conjecture.The reason is that we have to use formula (4.3) in a larger range of z when truncating the integraland the estimate (4.23) in Lemma 4.8 is not enough when ϑ F > 0. We avoid these obstacles byprobabilistic methods in large deviation theory (see Cram´er [5], Hwang [18])). We consider theassociated distribution ν T, F , x (see below) and use the Selberg-Beurling functions to approximateregions in R r . The associated distribution ν T, F , x plays the role of localizing the measure by changing x . Eventually, we choose a suitable x in the proof of Propositions 2.1, 2.2, and the x is the saddle That there are some incomplete arguments when truncating the integral in his proof, but we can fix his proof by anontrivial modification. point of the integrand in Lemma 4.9. The Selberg-Beurling formula has the advantage that it onlyinvolves finite integrals.We introduce some notation. Define the R r -valued function F θ ,X ( t ) by F θ ,X ( t ) = (Re e − iθ P F ( + it, X ) , . . . , Re e − iθ r P F r ( + it, X )) , and µ T, F the measure on R r by µ T, F ( B ) := T meas( F − θ ,X ( B ) ∩A ) for B ∈ B ( R r ). Put y j = V j σ F j ( X ).Then we find that1 T meas( S X ( T, V ; F , θ )) (4.28)= µ T, F (( y , ∞ ) × · · · × ( y r , ∞ )) + O F (cid:16) exp (cid:16) − b (log log X ) r +1) (cid:17)(cid:17) since meas([ T, T ] \ A ) ≪ F T exp( − b (log log X ) r +1) ) by Lemma 4.1. For x = ( x , . . . , x r ) ∈ R r , set ν T, F , x ( B ) := Z B e x ξ + ··· + x r ξ r dµ T, F ( ξ )for B ∈ B ( R r ). Note that ν T, F , x is a measure on R r , and has a finite value for every B ∈ B ( R r ), x ∈ R r , X ≥ ν T, F , x ( B ) ≤ ν T, F , x ( R r ) = 1 T Z A exp r X j =1 x j Re e − iθ j P F j ( + it, X ) dt < + ∞ . This is an analogue of the associated distribution function of ˜ M n ( w ) in Hwang [18, page 300]. Underthe above notation, we state and prove three lemmas. Lemma 4.9. For x , . . . , x r > , we have µ T, F (( y , ∞ ) × · · · × ( y r , ∞ ))= Z ∞ x r y r · · · Z ∞ x y e − ( τ + ··· + τ r ) ν T, F , x (( y , τ /x ) × · · · × ( y r , τ r /x r )) dτ · · · dτ r . Proof. For every B ∈ B ( R r ), it holds that µ T, F ( B ) = Z B e − ( x v + ··· + x r v r ) dν T, F , x ( v ) . By Fubini’s theorem, we find that Z ( y , ∞ ) ×···× ( y r , ∞ ) e − ( x v + ··· + x r v r ) dν T, F , x ( v )= Z ( y , ∞ ) ×···× ( y r , ∞ ) (cid:18)Z ∞ x r v r · · · Z ∞ x v e − ( τ + ··· + τ r ) dτ · · · dτ r (cid:19) dν T, F , x ( v )= Z ∞ x r y r · · · Z ∞ x y e − ( τ + ··· + τ r ) Z ( y ,τ /x ) ×···× ( y r ,τ r /x r ) dν T, F , x ( v ) ! dτ · · · dτ r = Z ∞ x r y r · · · Z ∞ x y e − ( τ + ··· + τ r ) ν T, F , x (( y , τ /x ) × · · · × ( y r , τ r /x r )) dτ · · · dτ r . (cid:3) The next lemma is a generalization of [23, Lemma 6.2] in multidimensions. Define G ( u ) = 2 uπ + 2(1 − u ) u tan πu , f c,d ( u ) = e − πicu − e − πidu . For a set A , we denote the indicator function of A by A . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 37 Lemma 4.10. Let L be a positive number. Let c , . . . , c r , d , . . . , d r be real numbers with c j < d j .Put R = ( c , d ) × · · · × ( c r , d r ) ⊂ R r . For any ξ = ( ξ , . . . , ξ r ) ∈ R r , we have R ( ξ ) = W L, R ( ξ ) + O r r X j =1 ((cid:18) sin( πL ( ξ j − c j )) πL ( ξ j − c j ) (cid:19) + (cid:18) sin( πL ( ξ j − d j )) πL ( ξ j − d j ) (cid:19) ) , where W L, R ( ξ ) is defined as if r is even, i r r − r X j =1 ( − j − Re r Y h =1 Z L G (cid:16) uL (cid:17) e πiε j ( h ) uξ h f c h ,d h ( ε j ( h ) u ) duu , if r is odd, i r +1 r − r X j =1 ( − j − Im r Y h =1 Z L G (cid:16) uL (cid:17) e πiε j ( h ) uξ h f c h ,d h ( ε j ( h ) u ) duu . Here, ε j ( h ) = 1 if ≤ h ≤ j − , and ε j ( h ) = − otherwise.Proof. We use the following formula (cf. [23, equation (6.1)]) ( c h ,d h ) ( ξ h ) = Im Z L G (cid:16) uL (cid:17) e πiuξ h f c h ,d h ( u ) duu + O (cid:18) sin( πL ( ξ h − c h )) πL ( ξ h − c h ) (cid:19) + (cid:18) sin( πL ( ξ h − d h )) πL ( ξ h − d h ) (cid:19) ! , which leads to the estimate Im R L G (cid:0) uL (cid:1) e πiuξ j f c j ,d j ( u ) duu ≪ 1. Therefore, we obtain R ( ξ ) = r Y h =1 Im Z L G (cid:16) uL (cid:17) e πiuξ h f c h ,d h ( u ) duu (4.29)+ O r r X j =1 ((cid:18) sin( πL ( ξ j − c j )) πL ( ξ j − c j ) (cid:19) + (cid:18) sin( πL ( ξ j − d j )) πL ( ξ j − d j ) (cid:19) ) . For any complex numbers w , . . . , w r , we observe thatIm( w ) · · · Im( w r ) = i r r r X j =1 ( − j − (cid:0) w · · · w j − w j · · · w r + ( − r w · · · w j · · · w r (cid:1) . In particular, if r is even, thenIm( w ) · · · Im( w r ) = i r r − Re r X j =1 ( − j − w · · · w j − w j · · · w r , and if r is odd, thenIm( w ) · · · Im( w r ) = i r +1 r − Im r X j =1 ( − j − w · · · w j − w j · · · w r . Substituting these to (4.29), we obtain Lemma 4.10. (cid:3) Lemma 4.11. Assume that F , θ satisfy (S4), (A1), and (A2). Let c , . . . , c r , d , . . . , d r be realnumbers with c j < d j . Put R = ( c , d ) ×· · ·× ( c r , d r ) . Let T , X be large numbers depending on F andsatisfying X (log log X ) r +1) ≤ T . Then for any x = ( x , . . . , x r ) ∈ R r satisfying k x k ≤ (log log X ) r ,we have ν T, F , x ( R ) = Ξ X ( x ) r Y h =1 e x h σ Fh ( X ) ! × r Y j =1 Z x j σ Fj ( X ) − cjσFj ( X ) x j σ Fj ( X ) − djσFj ( X ) e − v / dv √ π + E , (4.30) where the error term E satisfies E ≪ F exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) (log log X ) α F + + exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) √ log log X r Y h =1 d h − c h σ F h ( X ) for some constant C = C ( F ) > . Moreover, if k x k ≤ b with b = b ( F ) > sufficiently small, wehave ν T, F , x ( R ) = r Y h =1 e x h σ Fh ( X ) ! × r Y j =1 Z x j σ Fj ( X ) − cjσFj ( X ) x j σ Fj ( X ) − djσFj ( X ) e − v / dv √ π + E , (4.31) where the error term E satisfies E ≪ F X ) α F + + r X k =1 (cid:18) x k ( d k − c k ) σ F k ( X ) + 1 σ F k ( X ) (cid:19) r Y h =1 h = k d h − c h σ F h ( X ) . Proof. We show formula (4.30). Put L = b (log log X ) α F with b = b ( F ) a small positive constantto be chosen later. Recall that α F = min n r, − ϑ F ϑ F o . It follows from Lemma 4.10 that ν T, F , x ( R ) = Z R r W L, R ( ξ ) e x ξ + ··· + x r ξ r dµ T, F ( ξ ) + E, (4.32)where the error term E satisfies the estimate E ≪ r X j =1 Z R r ((cid:18) sin( πL ( ξ j − c j )) πL ( ξ j − c j ) (cid:19) + (cid:18) sin( πL ( ξ j − d j )) πL ( ξ j − d j ) (cid:19) ) e x ξ + ··· + x r ξ r dµ T, F ( ξ ) . First, we estimate E . For z = ( z , . . . , z r ) ∈ C r , define˜ M T ( z ) = Z R r e z ξ + ··· + z r ξ r dµ T, F ( ξ ) . Then, it holds that ˜ M T ( z ) = 1 T Z A exp r X j =1 z j Re e − iθ j P F j ( + it, X ) dt. Put w = ( w , . . . , w r ) = ( x + iu , . . . , x r + iu r ) with u j ∈ R . When k ( u , . . . , u r ) k ≤ L holds, wehave | ˜ M T ( w ) |≤ (cid:12)(cid:12)(cid:12)(cid:12) r Y j =1 exp (cid:18) ( x j + iu j ) (cid:0) σ F j ( X ) − σ F j ( | w j | ) (cid:1) + O F (cid:18) | x j + iu j | + | x j + iu j | − ϑ F − ϑ F (cid:19)(cid:19) (cid:12)(cid:12)(cid:12)(cid:12) + O F (cid:16) exp (cid:16) − b (log log X ) r +1) (cid:17)(cid:17) ≪ F exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) r Y j =1 exp x j (cid:0) σ F j ( X ) − σ F j ( | w j | ) (cid:1) − u j σ F j ( X ) + O F (cid:18) u j L ϑ F − ϑ F (cid:19)! by Proposition 4.1 and (4.23), where C = C ( F ) is some positive constant. Additionally, by (4.22), wefind that r Y j =1 exp − x j σ F j ( | w j | ) ! ≤ r Y j =1 exp − x j σ F j ( | x j | ) ! ≪ F Ξ X ( x ) exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) OINT VALUE DISTRIBUTION OF L -FUNCTIONS 39 for some C = C ( F ) > 0. By the definition of α F , the inequality L ϑ F − ϑ F ≤ (2 b ) ϑ F − ϑ F log log X holds.Therefore, when b is sufficiently small, we have | ˜ M T ( x + iu , . . . , x r + iu r ) | (4.33) ≪ F Ξ X ( x ) exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) r Y j =1 exp x j − u j ! σ F j ( X ) ! for k ( u , . . . , u r ) k ≤ L . For any ℓ, ξ ∈ R , we can write (cid:18) sin( πL ( ξ − ℓ )) πL ( ξ − ℓ ) (cid:19) = 2 L Z L ( L − u ) cos(2 π ( ξ − ℓ ) u ) du = 2 L Re Z L ( L − u ) e πi ( ξ − ℓ ) u du. (4.34)Thus Z R r (cid:18) sin( πL ( ξ j − ℓ )) πL ( ξ j − ℓ ) (cid:19) e x ξ + ··· + x r ξ r dµ T, F ( ξ )= 2 L Re Z L ( L − u ) Z R r e πi ( ξ j − ℓ ) u e x ξ + ··· + x r ξ r dµ T, F ( ξ ) du = 2 L Re Z L e − πiℓu ( L − u ) ˜ M T ( x , . . . , x j − , x j + 2 πiu, x j +1 , . . . , x r ) du, which, by (4.33), is ≪ F Ξ X ( x ) exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) L r Y k =1 exp (cid:18) x k σ F k ( X ) (cid:19)! Z L ( L − u ) exp (cid:0) − ( πσ F j ( X ) u ) (cid:1) du ≪ Ξ X ( x ) exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) Lσ F j ( X ) r Y k =1 exp (cid:18) x k σ F k ( X ) (cid:19) . It then follows that equation (4.32) satisfies ν T, F , x ( R ) = Z R r W L, R ( ξ ) e x ξ + ··· + x r ξ r dµ T, F ( ξ ) + E (4.35)with E ≪ F Ξ X ( x ) exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) L √ log log X r Y k =1 exp (cid:18) x k σ F k ( X ) (cid:19) . For the main term in (4.35), it is enough to calculate Z R r r Y h =1 Z L G (cid:16) uL (cid:17) e πiε j ( h ) uξ h f c h ,d h ( ε j ( h ) u ) duu ! e x ξ + ··· + x r ξ r dµ T, F ( ξ ) (4.36)for every fixed 1 ≤ j ≤ r . Using Fubini’s theorem, we find that (4.36) is equal to Z L · · · Z L r Y h =1 G (cid:16) u h L (cid:17) f c h ,d h ( ε j ( h ) u h ) u h ! × ˜ M T ( x + 2 πiε j (1) u , . . . , x r + 2 πiε j ( r ) u r ) du · · · du r . Here we divide the range of this integral as Z L · · · Z L = Z · · · Z + r − X k =0 Z · · · Z D k , where Z · · · Z D k = Z · · · Z Z L k z }| {Z L · · · Z L . By estimate (4.33) and the estimates f c,d ( ± u ) u ≪ d − c , G ( u/L ) ≪ ≤ u ≤ L , the integral over D r − k for 1 ≤ k ≤ r is ≪ F Ξ X ( x ) exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) k − Y h =1 ( d h − c h ) Z exp (cid:18)(cid:18) x h − ( πu ) (cid:19) σ F h ( X ) (cid:19) du ! × ( d k − c k ) Z L exp (cid:18)(cid:18) x k − ( πu ) (cid:19) σ F k ( X ) (cid:19) du × r Y h = k +1 ( d h − c h ) Z L exp (cid:18)(cid:18) x h − ( πu ) (cid:19) σ F h ( X ) (cid:19) du ! ≪ F Ξ X ( x ) exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) e − σ Fk ( X ) r Y h =1 d h − c h σ F h ( X ) exp (cid:18) x h σ F h ( X ) (cid:19) . Hence, integral (4.36) is equal to Z · · · Z r Y h =1 G (cid:16) u h L (cid:17) f c h ,d h ( ε j ( h ) u h ) u h ! ˜ M T ( x + 2 πiε j (1) u , . . . , x r + 2 πiε j ( r ) u r ) du · · · du r + O F r X k =1 Ξ X ( x ) exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) e − σ Fk ( X ) r Y h =1 d h − c h σ F h ( X ) exp (cid:18) x h σ F h ( X ) (cid:19)! . (4.37)When k ( u , . . . , u r ) k ≤ 1, it follows from Proposition 4.1 and equation (4.25) that˜ M T ( x + 2 πiε j (1) u , . . . , x r + 2 πiε j ( r ) u r )= Ξ X ( x ) r Y h =1 O F | u h | exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) + x j + u j log X !! × exp (cid:18) x h + 4 πiε j ( h ) x h u h − π u h σ F h ( X ) (cid:19) + O F (cid:16) exp (cid:16) − b (log log X ) r +1) (cid:17)(cid:17) . Therefore, the integral of (4.37) is equal toΞ X ( x ) r Y h =1 Z (cid:18) O F (cid:18) u exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) + x h + u h log X (cid:19)(cid:19) G (cid:16) uL (cid:17) f c h ,d h ( ε j ( h ) u ) × exp (cid:18) x h + 4 πiε j ( h ) x h u − π u σ F h ( X ) (cid:19) duu + O F exp (cid:16) − c (log log X ) r +1) (cid:17) r Y h =1 Z (cid:12)(cid:12)(cid:12)(cid:12) G (cid:16) uL (cid:17) f c h ,d h ( u ) u (cid:12)(cid:12)(cid:12)(cid:12) du ! . (4.38) OINT VALUE DISTRIBUTION OF L -FUNCTIONS 41 Since G ( u/L ) ≪ f ch,dh ( ± u ) u ≪ d h − c h , we find that Z (cid:18) u exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) + x h + u log X (cid:19) G (cid:16) uL (cid:17) f c h ,d h ( ε j ( h ) u ) × exp (cid:18) x h + 4 πiε j ( h ) x h u − π u σ F h ( X ) (cid:19) duu ≪ exp (cid:18) x h σ F h ( X ) (cid:19) ( d h − c h ) × Z (cid:18) u exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) + x h + u log X (cid:19) exp (cid:0) − π u σ F h ( X ) (cid:1) du ≪ F exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) × exp (cid:18) x h σ F h ( X ) (cid:19) d h − c h σ F h ( X ) , and that Z G (cid:16) uL (cid:17) f c h ,d h ( ε j ( h ) u ) exp (cid:18) x h + 4 πiε j ( h ) x h u − π u σ F h ( X ) (cid:19) duu ≪ ( d h − c h ) exp (cid:18) x h σ F h ( X ) (cid:19) Z exp (cid:0) − π u σ F h ( X ) (cid:1) du ≪ exp (cid:18) x h σ F h ( X ) (cid:19) d h − c h σ F h ( X ) . Moreover, we find that Z L G (cid:16) uL (cid:17) f c h ,d h ( ε j ( h ) u ) exp (cid:18) x h + 4 πiε j ( h ) x h u − π u σ F h ( X ) (cid:19) duu ≪ F exp (cid:18) x h σ F h ( X ) (cid:19) d h − c h σ F h ( X ) e − σ Fh ( X ) , and that Z (cid:12)(cid:12)(cid:12)(cid:12) G (cid:16) uL (cid:17) f c h ,d h ( ε j ( h ) u ) u (cid:12)(cid:12)(cid:12)(cid:12) du ≪ d h − c h . Also, by (4.22), it holds thatΞ X ( x ) exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) r Y j =1 exp x j σ F j ( X ) ! × √ log log X r Y h =1 σ F h ( X ) ≥ √ log log X r Y h =1 σ F h ( X ) ≥ exp (cid:16) − b (log log X ) r +1) (cid:17) for some constant D = D ( F ) > 0. From these estimates and (4.38), integral (4.37) is equal toΞ X ( x ) r Y h =1 Z L G (cid:16) uL (cid:17) f c h ,d h ( ε j ( h ) u ) exp (cid:18) x h + 4 πiε j ( h ) x h u − π u σ F h ( X ) (cid:19) duu + O F Ξ X ( x ) exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) r Y j =1 exp x j σ F j ( X ) ! × √ log log X r Y h =1 d h − c h σ F h ( X ) with D = max { D , D } . Using the well known formula1 √ π Z R e − ivξ e − ηv dv = 1 √ η exp (cid:18) − ξ η (cid:19) , (4.39) we can rewrite the above main term asΞ X ( x ) r Y h =1 exp (cid:16) x h σ F h ( X ) (cid:17) √ π × Z R r e − ( v + ··· + v r ) / (cid:26) r Y h =1 Z L G (cid:16) uL (cid:17) e πiε j ( h ) u ( x h σ Fh ( X ) − v h σ Fh ( X )) f c h ,d h ( ε j ( h ) u ) duu (cid:27) d v . Combining this with (4.37), we see that integral (4.36) is equal toΞ X ( x ) r Y h =1 exp (cid:16) x h σ F h ( X ) (cid:17) √ π × Z R r e − ( v + ··· + v r ) / (cid:26) r Y h =1 Z L G (cid:16) uL (cid:17) e πiε j ( h ) u ( x h σ Fh ( X ) − v h σ Fh ( X )) f c h ,d h ( ε j ( h ) u ) duu (cid:27) d v + O F Ξ X ( x ) exp (cid:18) D k x k − ϑ F − ϑ F (cid:19) r Y j =1 exp x j σ F j ( X ) ! × √ log log X r Y h =1 d h − c h σ F h ( X ) . Substituting this equation to the definition of W L, R and using Lemma 4.10 and equation (4.35), weobtain ν T, F , x ( R )= Ξ X ( x ) r Y h =1 exp (cid:16) x h σ F h ( X ) (cid:17) √ π × (cid:26) Z R r e − v ··· + v r R (cid:0) x σ F ( X ) − v σ F ( X ) , . . . , x r σ F r ( X ) − v r σ F r ( X ) (cid:1) d v + E + E (cid:27) , where E and E satisfy E ≪ F r X j =1 Z R r (cid:26) (cid:18) sin( πL ( x j σ F j ( X ) − v j σ F j ( X ) − c j )) πL ( x j σ F j ( X ) − v j σ F j ( X ) − c j ) (cid:19) + (cid:18) sin( πL ( x j σ F j ( X ) − v j σ F j ( X ) − d j )) πL ( x j σ F j ( X ) − v j σ F j ( X ) − d j ) (cid:19) (cid:27) × e − ( v + ··· + v r ) / d v , and E ≪ F exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) (log log X ) α F + + exp (cid:18) C k x k − ϑ F − ϑ F (cid:19) √ log log X r Y h =1 d h − c h σ F h ( X )for some constant C = C ( F ) > 0. By equation (4.34), it holds that, for any ℓ ∈ R , Z R r (cid:18) sin( πL ( x j σ F j ( X ) − v j σ F j ( X ) − ℓ )) πL ( x j σ F j ( X ) − v j σ F j ( X ) − ℓ ) (cid:19) × e − ( v + ··· + v r ) / d v = 2 L Re Z L ( L − α ) (cid:18)Z R r e πi ( x j σ Fj ( X ) − v j σ Fj ( X ) − ℓ ) α e − ( v + ··· + v r ) / d v (cid:19) dα = 2(2 π ) ( r − / L Re Z L ( L − α ) e πi ( x j σ Fj ( X ) − ℓ ) α (cid:18)Z R e − πivσ Fj ( X ) α e − v / dv (cid:19) dα, OINT VALUE DISTRIBUTION OF L -FUNCTIONS 43 which, by (4.39), becomes= 2(2 π ) r/ L Re Z L ( L − α ) e πi ( x j σ Fj ( X ) − ℓ ) α exp (cid:0) − π α σ F j ( X ) (cid:1) dα ≪ F Lσ F j ( X ) ≪ F X ) α F + . Hence, we have E ≪ F X ) α F + 12 . Finally, by simple calculations, we can write Z R r e − v ··· + v r R (cid:0) x σ F ( X ) − v σ F ( X ) , . . . , x r σ F r ( X ) − v r σ F r ( X ) (cid:1) d v = r Y j =1 Z x j σ Fj ( X ) − cjσFj ( X ) x j σ Fj ( X ) − djσFj ( X ) e − v / dv √ π and this completes the proof of (4.30).Next, we consider (4.31). Using Proposition 4.1 and equation (4.24), we have M T ( x + 2 πiε j (1) u , . . . , x r + 2 πiε j ( r ) u r )= r Y h =1 (cid:0) O F (cid:0) | x h + iu h | (cid:1)(cid:1) exp (cid:18) x h + 4 πiε j ( h ) x h u h − π u h σ F h ( X ) (cid:19) + O F (cid:16) exp (cid:16) − b (log log X ) r +1) (cid:17)(cid:17) when k x k , k u k are sufficiently small. By using this equation, we can prove (4.31) similarly to theproof of (4.30). (cid:3) Proof of Proposition 2.1. We firstly prove Proposition 2.1 in the case V j ’s are nonnegative. Let x =( x , . . . , x r ) ∈ ( R > ) r satisfying k x k ≤ b with b the same number as in Lemma 4.11. By Lemma 4.9and equation (4.31), we have µ T, F (( y , ∞ ) × · · · × ( y r , ∞ ))= r Y j =1 e x j σ Fj ( X ) Z ∞ x j y j e − τ Z x j σ Fj ( X ) − yjσFj ( X ) x j σ Fj ( X ) − τ/xjσFj ( X ) e − v / dv √ π dτ + O F X ) α F + r Y j =1 e x j σ Fj ( X ) − x j y j + E × r Y j =1 e x j σ Fj ( X ) , where E = r X k =1 Z ∞ x r y r · · · Z ∞ x y e − ( τ + ··· + τ r ) (cid:18) x k ( τ k /x k − y k ) σ F k ( X ) + 1 σ F k ( X ) (cid:19) r Y h =1 h = k τ h /x h − y h σ F h ( X ) dτ · · · dτ r . Now, simple calculations lead that Z ∞ x j y j e − τ Z x j σ Fj ( X ) − yjσFj ( X ) x j σ Fj ( X ) − τ/xjσFj ( X ) e − v / dv √ π dτ = exp − x j σ F j ( X ) ! Z ∞ V j e − u / du √ π since y j = V j σ F j ( X ). Therefore, we obtain µ T, F (( y , ∞ ) × · · · × ( y r , ∞ ))= r Y j =1 Z ∞ V j e − u / du √ π + O F X ) α F + r Y j =1 e x j σ Fj ( X ) − x j y j + E r Y j =1 e x j σ Fj ( X ) . Here, we decide x j ’s as x j = max { , V j } /σ F j ( X ), where V j ’s must satisfy the inequality V j ≤ Rσ F j ( X ).Then, we see that e − x j y j = e − x j σ Fj ( X ) e x j σ Fj ( X ) − x j y j ≪ e − x j σ Fj ( X ) e − V j / . (4.40)This estimate leads that1(log log X ) α F + r Y j =1 e x j σ Fj ( X ) − x j y j ≪ r e − ( V + ··· + V r ) / (log log X ) α F + ≪ r X ) α F + r Y j =1 (1 + V j ) Z ∞ V j e − u / du √ π . Moreover, since it holds that Z ∞ x j y j (cid:18) τx j − y j (cid:19) e − τ dτ = e − x j y j x j , (4.41)we have Z ∞ x r y r · · · Z ∞ x y e − ( τ + ··· + τ r ) (cid:18) x k ( d k − c k ) σ F k ( X ) + 1 σ F k ( X ) (cid:19) r Y h =1 h = k τ h /x h − y h σ F h ( X ) dτ · · · dτ r = x k r Y j =1 σ F j ( X ) Z ∞ x j y j (cid:18) τx j − y j (cid:19) e − τ dτ + e − x k y k σ F k ( X ) r Y j =1 j = k σ F j ( X ) Z ∞ x j y j (cid:18) τx j − y j (cid:19) e − τ dτ = x k r Y j =1 e − x j y j x j σ F j ( X ) + x k σ F k ( X ) r Y j =1 e − x j y j x j σ F j ( X ) . (4.42)for every 1 ≤ k ≤ r . By estimate (4.40) and x j σ F j ( X ) ≍ V j , we can write e − x j y j x j σ F j ( X ) ≪ e − x j σ Fj ( X ) e − V j / V j ≪ e − x j σ Fj ( X ) Z ∞ V j e − u / du √ π . By this observation, (4.42) is ≪ r (cid:18) x k + x k σ F k ( X ) (cid:19) r Y j =1 e − x j σ Fj ( X ) Z ∞ V j e − u / du √ π ≪ F V k log log X r Y j =1 e − x j σ Fj ( X ) Z ∞ V j e − u / du √ π . Hence, we have E r Y j =1 e x j σ Fj ( X ) ≪ F k V k log log X r Y j =1 Z ∞ V j e − u / du √ π . From the above estimations, we obtain µ T, F (( y , ∞ ) × · · · × ( y r , ∞ ))= r Y j =1 Z ∞ V j e − u / du √ π + O F (cid:26) Q rk =1 (1 + V k )(log log X ) α F + + 1 + k V k log log X (cid:27) × r Y j =1 Z ∞ V j e − u / du √ π for 0 ≤ V j ≤ b σ F j ( X ). Thus, by this formula and (4.28), we complete the proof of Proposition 2.1 inthe case V j ’s are nonnegative. OINT VALUE DISTRIBUTION OF L -FUNCTIONS 45 In order to finish the proof of Proposition 2.1, we consider the negative cases. It suffices to showthat, for the case − bσ F ( X ) ≤ V ≤ ≤ V j ≤ bσ F j ( X ),1 T meas( S X ( T, ( − V , V , . . . , V r ); F , θ ))= (cid:18) O F (cid:18) Q rk =1 (1 + | V k | )(log log X ) α F + + 1 + k V k log log X (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u du √ π since other cases can be shown similarly by induction. By the definition of the set S X ( T, V ; F , θ ), itholds that S X ( T, V ; F , θ ) = S X ( T, ( V , . . . , V r ); ( F , . . . , F r ) , ( θ , . . . , θ r )) \ S X ( T, ( − V − , V , . . . , V r ); F , ( π − θ , θ , . . . , θ r )) , where we regard that if r = 1, the first set on the right hand side is [ T, T ]. Therefore, from thenonnegative cases, we have1 T meas( S X ( T, ( − V , V , . . . , V r ); F , θ )) (4.43)= 1 T meas( S X ( T, ( V , . . . , V r ); ( F , . . . , F r ) , ( θ , . . . , θ r ))) − T meas( S X ( T, ( − V − , V , . . . , V r ); F , ( π − θ , θ , . . . , θ r )))= (1 + E ) r Y j =2 Z ∞ V j e − u du √ π − (1 + E ) r Y j =1 Z ∞| V j | e − u du √ π . Here, E and E satisfy E ≪ F Q rk =2 (1 + V k )(log log X ) α F + + 1 + k ( V , . . . , V r ) k log log X ,E ≪ F Q rk =1 (1 + V k )(log log X ) α F + + 1 + k ( V , . . . , V r ) k log log X . Hence, we find that (4.43) is equal to (cid:18) − Z ∞− V e − u / du √ π (cid:19) r Y j =2 Z ∞ V j e − u / du √ π + E r Y j =2 Z ∞ V j e − u du √ π + E r Y j =1 Z ∞ V j e − u du √ π = (cid:18) O F (cid:18) Q rk =1 (1 + | V k | )(log log X ) α F + + 1 + k V k log log X (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u / du √ π . Thus, we also obtain the negative cases of Proposition 2.1. (cid:3) Proof of Proposition 2.2. Let V = ( V , . . . , V r ) ∈ ( R ≥ ) r satisfying k V k ≤ (log log X ) r , and put x j = max { , V j } /σ F j ( X ). Similarly to the proof of Proposition 2.1 by using (4.30) instead of (4.31), we obtain µ T, F (( y , ∞ ) × · · · × ( y r , ∞ ))= Ξ X ( x ) ( r Y j =1 Z ∞ V j e − u / du √ π + O F exp C (cid:18) k V k√ log log X (cid:19) − ϑ F − ϑ F ! Q rk =1 (1 + V k )(log log X ) α F + r Y j =1 Z ∞ V j e − u / du √ π + E ) for 0 ≤ V j ≤ (log log X ) r , where E = exp C (cid:18) k V k√ log log X (cid:19) − ϑ F − ϑ F √ log log X × r Y j =1 e x j σ Fj ( X ) σ F j ( X ) Z y j x j (cid:18) τx j − y j (cid:19) e − τ dτ. Here, C = C ( F ) is a positive constant. Moreover, using (4.41) we have E = exp C (cid:18) k V k√ log log X (cid:19) − ϑ F − ϑ F √ log log X r Y j =1 e x j σ Fj ( X ) − x j y j x j σ F j ( X )By estimate (4.40) and x j σ F j ( X ) ≍ V j , we can write e x j σ Fj ( X ) − x j y j x j σ F j ( X ) ≪ e − V j / V j ≪ Z ∞ V j e − u / du √ π . By this observation, we have E ≪ r exp C (cid:18) k V k√ log log X (cid:19) − ϑ F − ϑ F √ log log X r Y j =1 Z ∞ V j e − u / du √ π . From the above estimations, we obtain µ T, F (( y , ∞ ) × · · · × ( y r , ∞ ))= Ξ X ( x ) r Y j =1 Z ∞ V j e − u / du √ π × ( O F exp C (cid:18) k V k√ log log X (cid:19) − ϑ F − ϑ F (cid:26) Q rk =1 (1 + V k )(log log X ) α F + + 1 √ log log X (cid:27) ) . In particular, by the definition of Ξ X (2.20), assumptions (A1), (A2), and the analyticity of Ξ X comingfrom Lemma 4.6, it holds thatΞ X ( x ) = (cid:18) O F (cid:18) √ log log X (cid:19)(cid:19) Ξ X (cid:16) V σ F ( X ) , . . . , V r σ Fr ( X ) (cid:17) . Thus, by these formulas and (4.28), we obtain Proposition 2.2 when k V k ≤ (log log X ) r . (cid:3) Proof of Theorem 2.6. We prove the following lemma as a preparation. Lemma 4.12. Let F be a Dirichlet series satisfying (S4), (S5), and (2.1) . There exists a positiveconstant c = c ( F ) such that for any large numbers X , T with X ≤ T / and for any ∆ > , σ F ( X ) + ∆ − ≤ V ≤ log ( T/ log T )∆ log X T meas (cid:8) t ∈ [ T, T ] : P F ( + it, X ) > V (cid:9) ≪ exp (cid:18) − ∆ V log (cid:18) c V ∆ σ F ( X ) (cid:19)(cid:19) . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 47 Proof. Similarly to the proof of (4.2), we can obtain1 T Z TT | P F ( + it, X ) | k dt ≪ ( Ckσ F ( X ) ) k for any 1 ≤ k ≤ log ( T/ log T )log X with C = C ( F ) a positive constant. It follows from this estimate that1 T meas (cid:8) t ∈ [ T, T ] : P F ( + it, X ) > V (cid:9) ≪ (cid:18) Ckσ F ( X ) V (cid:19) k . Hence, choosing k = ⌊ ∆ V ⌋ , we obtain1 T meas (cid:8) t ∈ [ T, T ] : P F ( + it, X ) > V (cid:9) ≪ exp (cid:18) − ∆ V log (cid:18) V σ F ( X ) (cid:19)(cid:19) for any V ≤ log ( T/ log T )∆ log X . (cid:3) Proof of Theorem 2.6. To make the dependency of the limit on X and T explicit, we introduce threemore parameters n ∗ F , ϑ ∗ F , and b ∗ F , where n ∗ F = min ≤ j ≤ r n F j , ϑ ∗ F = min ≤ j ≤ r ϑ F j , b ∗ F = min ≤ i ≤ rϑ Fi = ϑ ∗ F sup p prime { | b F i ( p ) | p ϑ Fi } . Let ε > X ≤ max { c (log T log log T ) ϑ ∗ F , c T ) log log T log T } ,where c = c ( F , k, ε ) and c = c ( F , k, ε ) are some positive constants to be chosen later. Denote φ F ( t, X ) = min ≤ j ≤ r Re e − iθ j P F j ( + it, X ) andΦ F ( T, V, X ) = meas { t ∈ [ T, T ] : φ F ( t, X ) > V } . Then we have Z TT exp (2 kφ F ( t, X )) dt = Z ∞−∞ ke kV Φ F ( T, V, X ) dV. We divide the range of the integral on the right hand side as I + I + I + I + I := (cid:18) Z √ log log X −∞ + Z Ck log log X √ log log X + Z log T log log T Ck log log X + Z c log T log T log log T + Z ∞ c log T (cid:19) ke kV Φ F ( T, V, X ) dV with c = k + √ ε/ and C = C ( F ) a suitably large constant. The trivial bound Φ F ( T, V, X ) ≤ T yields the inequality I ≤ T e k √ log log X . Applying Lemma 4.12 with ∆ = 4 k , we have I ≪ T when C is suitably large depending only on F . Similarly, applying Lemma 4.12 with ∆ = k + √ ε/ T , wehave I ≪ k,ε T .Next, we consider I . We have φ F ( t, X ) ≤ min ≤ j ≤ r | P F j ( + it, X ) | ≤ min ≤ j ≤ r X p ≤ X | a F j ( p ) | p / + max ≤ j ≤ r X ≤ ℓ ≤ log X log 2 X p ≤ X /ℓ | b F j ( p ℓ ) | p ℓ/ . By the Cauchy-Schwarz inequality, the prime number theorem, and assumption (2.1), the first sum is ≤ min ≤ j ≤ r X p ≤ X / X p ≤ X | a F j ( p ) | p / = (1 + o (1)) s X log X × n ∗ F log log X. On the other hand, we also obtain, using the prime number theorem and partial summation, that thefirst sum is ≤ min ≤ j ≤ r sup p | b F j ( p ) | p ϑ Fj X p ≤ X p ϑ Fj p / = (1 + o (1)) 2 b ∗ F ϑ ∗ F X / ϑ ∗ F log X By the assumption (S5) and the Cauchy-Schwarz inequality, the second sum is ≪ F j X ≤ ℓ ≤ log X log 2 X p ≤ X /ℓ p ℓ ) / X p ≤ X /ℓ | b F j ( p ℓ ) log p ℓ | p ℓ / ≪ X / . Therefore, we have φ F ( t, X ) ≤ (1 + o (1)) min ( b ∗ F ϑ ∗ F X / ϑ ∗ F log X , s n ∗ F X log log X log X ) . Choosing c = ( b ∗ F (4 k + √ ε/ − / (1+2 ϑ ∗ F ) and c = ((8 k + ε ) n ∗ F ) − , we have Φ F ( T, V, X ) = 0 for V ≥ c log T and thus I = 0.Finally, we consider I . From Proposition 2.2, it holds thatΦ F ( T, V, X ) = T (1 + E ) × Ξ X (cid:16) Vσ F ( X ) , . . . , Vσ Fr ( X ) (cid:17) r Y j =1 Z ∞ V/σ Fj ( X ) e − u / du √ π for √ log log X ≤ V ≤ Ck log log X . Here, E satisfies the estimate E ≪ F ,k V r (log log X ) α F + r +12 + 1 √ log log X . We also have Z ∞ x e − u / du √ π = e − x / √ πx (cid:18) O ε (cid:18) x (cid:19)(cid:19) for x ≥ ε with any ε > 0. Therefore, we obtainΦ F ( T, V, X )= T r Y j =1 σ F j ( X ) E + O F (log log X/V )(2 π ) r/ V r × Ξ X (cid:16) Vσ F ( X ) , . . . , Vσ Fr ( X ) (cid:17) exp (cid:0) − H F ( X ) − V (cid:1) for √ log log X ≤ V ≤ Ck log log X . Hence, we have2 ke kV Φ F ( T, V, X ) (4.44)= T k r Y j =1 σ F j ( X ) E + O (log log X/V )(2 π ) r/ V r exp (cid:0) k H F ( X ) (cid:1) × Ξ X (cid:16) Vσ F ( X ) , . . . , Vσ Fr ( X ) (cid:17) exp (cid:18) − H F ( X ) ( V − kH F ( X )) (cid:19) for √ log log X ≤ V ≤ Ck log log X . Put H = D p log log X log X with D = D ( F ) a suitably largeconstant. Here, we choose X ( F , k ) large such that kH F ( X ) ≥ H if necessary. We write I = Z kH F ( X ) − H √ log log X + Z kH F ( X )+ HkH F ( X ) − H + Z Ck log log XkH F ( X )+ H ! ke kV Φ F ( T, V, X ) dV =: I , + I , + I , . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 49 Applying (4.44), we find that I , ≪ F ,k T exp (cid:0) k H F ( X ) (cid:1) Z kH F ( X ) − H √ log log X exp (cid:18) − H F ( X ) ( V − kH F ( X )) (cid:19) dV ≤ T exp (cid:0) k H F ( X ) (cid:1) Z ∞ H exp (cid:18) − V H F ( X ) (cid:19) dV ≪ T H F ( X ) H exp (cid:18) k H F ( X ) − H H F ( X ) (cid:19) , and that I , ≪ F ,k T exp( k H F ( X ))(log log X ) r/ Z Ck log log XkH F ( X )+ H exp (cid:18) − H F ( X ) ( V − kH F ( X )) (cid:19) dV ≤ T exp( k H F ( X ))(log log X ) r/ Z ∞ H exp (cid:18) − V H F ( X ) (cid:19) dV ≪ T H F ( X ) H (log log X ) r/ exp (cid:18) k H F ( X ) − H H F ( X ) (cid:19) . Hence, we have I , + I , ≪ F ,k T exp (cid:0) k H F ( X ) (cid:1) (log log X ) r Our main term comes from I , , which is equal to2 k exp (cid:0) k H F ( X ) (cid:1) ( √ πkH F ( X )) r r Y j =1 σ F j ( X ) O F X ) α F − r − + 1 √ log log X + HH F ( X ) !! × Ξ X (cid:18) kH F ( X ) + O ( H ) σ F ( X ) , . . . , kH F ( X ) + O ( H ) σ F r ( X ) (cid:19) Z kH F ( X )+ HkH F ( X ) − H exp (cid:18) − H F ( X ) ( V − kH F ( X )) (cid:19) . The analyticity of Ξ X yields thatΞ X (cid:18) kH F ( X ) + O ( H ) σ F ( X ) , . . . , kH F ( X ) + O ( H ) σ F r ( X ) (cid:19) = (cid:18) O k, F (cid:18) H log log X (cid:19)(cid:19) Ξ X (cid:18) kH F ( X ) σ F ( X ) , . . . , kH F ( X ) σ F r ( X ) (cid:19) . We also see that Z kH F ( X )+ HkH F ( X ) − H exp (cid:18) − H F ( X ) ( V − kH F ( X )) (cid:19) = p πH F ( X ) + O (cid:18)Z ∞ H exp (cid:18) − V H F ( X ) (cid:19)(cid:19) = p πH F ( X ) + O (cid:18) H F ( X ) H exp (cid:18) − H H F ( X ) (cid:19)(cid:19) . Therefore, we have I , = T exp (cid:0) k H F ( X ) (cid:1) Q rj =1 σ F j ( X ) (cid:0) √ πkH F ( X ) (cid:1) r − q H F ( X ) Ξ X (cid:18) kH F ( X ) σ F ( X ) , . . . , kH F ( X ) σ F r ( X ) (cid:19) (1 + E ) , where E ≪ F ,k X ) α F − r − + log X √ log log X . From the above estimates of I , I , and I , we obtain Theorem 2.6. (cid:3) Proof of Corollary 2.3. Assume the Selberg Orthonormality Conjecture, (2.22) and ϑ F < r +1 . Notethat the error term E in Theorem 2.6 is = o (1) as X → + ∞ when ϑ F < r +1 . Hence, it suffices toshow that for k > (cid:0) k H F ( X ) (cid:1) Q rj =1 σ F j ( X ) (cid:0) √ πkH F ( X ) (cid:1) r − q H F ( X ) Ξ X (cid:18) kH F ( X ) σ F ( X ) , . . . , kH F ( X ) σ F r ( X ) (cid:19) ∼ C ( F , k, θ ) (log X ) k /h F (log log X ) ( r − / 20 S. INOUE AND J. LI as X → + ∞ . Using (2.22), (S4), (S5), we find that σ F j ( X ) = n F j X + c j + o (1) (4.45)as X → + ∞ . Here, c j is a constant depending only on F . From this equation, we also see that σ F j ( X ) = (1 + o (1)) q n Fj log log X . Moreover, it holds that as X → + ∞ H F ( X ) = Q rj =1 ( n F j log log X + 2 c j ) P rh =1 Q rj =1 j = h ( n F j log log X + 2 c j ) + o (1) = log log Xh F + g F + o (1) (4.46)for a constant g F . Hence, we haveexp (cid:0) k H F ( X ) (cid:1) Q rj =1 σ F j ( X ) (cid:0) √ πkH F ( X ) (cid:1) r − q H F ( X ) ∼ C ( F , k ) (log X ) k /h F (log log X ) r − as X → + ∞ . Also, by (2.22), (S4), and (S5), we can prove that, for j = j , τ j ,j ( X ) = c j ,j + o (1)as X → + ∞ , where c j ,j is a constant depending only on F , θ . From this equation and the definitionof Ψ F (4.7), we can writeΞ X (cid:18) kH F ( X ) σ F ( X ) , . . . , kH F ( X ) σ F r ( X ) (cid:19) = (1 + o (1)) exp k X ≤ j Under the same situation as in Proposition 3.2. Let r ∈ Z ≥ be given. There existsa positive constant A = A ( F, r ) such that for X = T / (log log T ) r +1) , Y = T /k , k ∈ Z ≥ with k ≤ (log log T ) r +1) , T Z TT (cid:12)(cid:12)(cid:12)(cid:12) log F ( + it ) − P F ( + it, X ) − X | + it − ρ F |≤ Y log (( + it − ρ F ) log Y ) (cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ A k k k + A k k !(log T ) k , and T Z TT (cid:12)(cid:12)(cid:12)(cid:12) log F ( + it ) − P F ( + it, X ) (cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ A k k k + A k k !(log T ) k . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 51 Proof. By Proposition 3.2, it suffices to show that X X
Proof of Theorem 2.1 . We consider (2.4) and (2.5)–(2.6) separately. Proof of (2.4) . Let T be large. Put X = T / (log log T ) r +1) . Let A ≥ E j be E j := (cid:26) t ∈ [ T, T ] : (cid:12)(cid:12)(cid:12)(cid:12) log F j ( + it ) − P F j ( + it, X ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ L (cid:27) . From Lemma 5.1, we have meas( E j ) ≪ T L − kj A k ( k k + k k (log T ) k )for all j with A := A ( F ) = max ≤ j ≤ r A ( F j , r ) + 1, where A ( F j , r ) has the same meaning asin Lemma 5.1. Here the parameter L satisfying L ≥ (2 A log T ) / will be chosen later. Set k = ⌊L / /eA / ⌋ so that meas( E j ) ≪ T exp( − c L / ) for some c > 0. Therefore except on the set E := S rj =1 E j with measure O r ( T exp( − c L / )), we haveRe e − iθ j log F j ( + it ) = Re e − iθ j P F j ( + it, X ) + β j ( t ) L (5.2)with | β j ( t ) | ≤ j = 1 , . . . , r . By (5.2) and Proposition 2.1, the measure of t ∈ [ T, T ] \E suchthat Re e − iθ j log F j ( + it ) ≥ V j r n F j T , ∀ j = 1 , . . . , r (5.3)is at least (since β j ( t ) ≤ T O F Q rj =1 (1 + | V k | + L√ log log T )(log log T ) α F + + 1 + k V k + L log log T log log T !! × r Y j =1 Z ∞ σ Fj ( X ) − ( V j √ ( n Fj / 2) log log T + L ) e − u / du √ π (5.4)for L√ log log T , k V k ≤ c √ log log T with c sufficiently small. Similarly, the measure of t ∈ [ T, T ] \E suchthat (5.3) holds is at most T O F Q rk =1 (1 + | V k | + L√ log log T )(log log T ) α F + + 1 + k V k + L log log T log log T !! × r Y j =1 Z ∞ σ Fj ( X ) − ( V j √ ( n Fj / 2) log log T −L ) e − u / du √ π (5.5)for L√ log log T , k V k ≤ c √ log log T . By using equation (2.1), we find that σ F j ( X ) = r n F j T + O r,F j (cid:18) log T √ log log T (cid:19) , and so we also have σ F j ( X ) − = 1 q n Fj log log T (cid:18) O r,F j (cid:18) log T log log T (cid:19)(cid:19) . (5.6)Therefore, when | V j | ≤ q log log T log T , ( | V j | + 1) L ≤ B √ log log T with B > Z ∞ σ Fj ( X ) − ( V j √ ( n Fj / 2) log log T ±L ) e − u / du √ π = Z ∞ V j e − u / du √ π + Z V j V j + O r,Fj ( | V j | log3 T log log T + L√ log log T ) e − u / du √ π = Z ∞ V j e − u / du √ π + O r,F j ,B (cid:18)(cid:18) | V j | log T log log T + L√ log log T (cid:19) e − V j / (cid:19) = (cid:18) O r,F j ,B (cid:18) | V j | ( | V j | + 1) log T + L ( | V j | + 1) √ log log T log log T (cid:19)(cid:19) Z ∞ V j e − u / du √ π . Hence, choosing L = 2 c − r ( k V k + (log T ) ), we find that (5.4) and (5.5) become T (cid:18) O F ,B (cid:18) ( k V k + (log T ) )( k V k + 1) √ log log T + Q rk =1 (1 + | V k | )(log log T ) α F + (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u du √ π , and T exp( − c L / ) ≪ T exp( − r ( k V k + log T )) ≪ T T r Y j =1 Z ∞ V j e − u / du √ π when k V k ≤ c r − B (log log T ) / . Choosing B = Ac − r , we have1 T meas r \ j =1 t ∈ [ T, T ] : Re e − iθ j log F j (1 / it ) q n Fj log log T ≥ V j = (cid:18) O F ,A (cid:18) ( k V k + (log T ) )( k V k + 1) √ log log T + Q rk =1 (1 + | V k | )(log log T ) α F +1 / (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u du √ π for k V k ≤ A (log log T ) / . Thus, we complete the proof of (2.4). (cid:3) Proof of (2.5) and (2.6) . Let X = T / (log log T ) r +1) , Y = T /k and let B j be the set of t ∈ [ T, T ]such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log F j ( + it ) − P F j ( + it, X ) − X | / it − ρ Fj |≤ Y log(( + it − ρ F j ) log Y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ L . By Lemma 5.1, we know meas( B j ) ≤ T L − k A k ( k k + k k (log T ) k ) , where A = max ≤ j ≤ r A ( F j , r ) + 2 and A ( F j , r ) has the same meaning as in Lemma 5.1. By taking k = ⌊L / √ A e ⌋ , we have that meas( B j ) ≤ T exp( − c L ) for some c > L ≥ A log T .Therefore, it follows that for t ∈ [ T, T ] \ S rj =1 B j Re e − iθ j log F j ( + it )= Re e − iθ j P F j ( + it, X ) + X | / it − ρ Fj |≤ Y Re e − iθ j log(( + it − ρ F j ) log Y ) + β j ( t ) L OINT VALUE DISTRIBUTION OF L -FUNCTIONS 53 holds for all 1 ≤ j ≤ r with some | β j ( t ) | ≤ 1. Let C j be the set of t ∈ [ T, T ] such that X | / − it − ρ Fj |≤ Y Re e − θ j log(( + it − ρ F j ) log Y ) ≥ L . When θ j ∈ [ − π , π ] and | / it − ρ F j | ≤ Y , we find thatRe e − iθ j log(( + it − ρ F j ) log Y )= cos θ j log | ( + it − ρ F j ) log Y | + sin θ j arg(( + it − ρ F j ) log Y ) ≤ π. Hence, we have X | / it − ρ Fj |≤ Y Re e − iθ j log(( + it − ρ F j ) log Y ) ≤ π X | / it − ρ Fj |≤ Y , and thus by Lemma 3.11 meas( C j ) ≤ C k k k T L − k for some constant C = C ( F j ) > 0. By choosing k = ⌊L / √ Ce ⌋ , we have meas( C j ) ≤ T exp( − c L ) forsome c > 0. Now we have that the measure of t ∈ [ T, T ] \ S rj =1 ( B j ∪ C j ) such thatRe e − θ j log F j ( + it ) ≥ V j r n F j T is bounded by the measure of the set t ∈ [ T, T ] such thatRe e − iθ j P F j ( + it, X ) ≥ V j r n F j T − L . (5.7)From Proposition 2.1, we know (5.7) holds with measure T O F Q rk =1 (1 + | V k | + L√ log log T )(log log T ) α F + + 1 + k V k + L log log T log log T !! × r Y j =1 Z ∞ σ Fj ( X ) − ( V j √ ( n Fj / 2) log log T − L ) e − u / du √ π (5.8)for L√ log log T , k V k ≤ c √ log log T with c sufficiently small. Since we have meas( S rj =1 ( B j ∪ C j )) ≪ T exp( − c L ) for some c > 0, we choose L = c − r ( k V k + 2 A log T ) so that that (5.8) becomes T (cid:18) O F ,A (cid:18) ( k V k + log T )( k V k + 1) √ log log T + Q rk =1 (1 + | V k | )(log log T ) α F + (cid:19)(cid:19) r Y j =1 Z ∞ V j e − u du √ π , for k V k ≤ A (log log T ) / . This completes the proof of (2.5).The proof of (2.6) is similar by noting that when θ j ∈ [ π , π ] and | + it − ρ F j | ≤ Y ,Re e − iθ j log(( + it − ρ F j ) log Y )= cos θ j log | ( + it − ρ F j ) log Y | + sin θ j arg(( + it − ρ F j ) log Y ) ≥ − π. and thus the set of t ∈ [ T, T ] such that X | / it − ρ Fj |≤ Y Re e − θ j log(( + it − ρ F j ) log Y ) ≤ −L has measure bounded by C k k k T L − k for some constant C = C ( F j ). (cid:3) Proof of Theorem 2.2. Let X = T / (log log T ) r +1) . Let a = a ( F ) > V = ( V , . . . , V r ) ∈ ( R ≥ ) r such that k V k ≤ a (1 + V / m )(log log T ) / with V m := min ≤ j ≤ r V j .We consider the case when θ ∈ [ − π , π ] r first. Similarly to the proof of (2.5) (see (5.7)), we findthat the measure of the set of t ∈ [ T, T ] except for a set of measure T exp( − c L ) ( L ≫ log T ) suchthat Re e − iθ j log F j (cid:0) + it (cid:1) ≥ V j q n Fj log log T is at most the measure of the set t ∈ [ T, T ] such thatRe e − iθ j P F j ( + it, X ) ≥ V j r n F j T − L . From Proposition 2.1, the measure of t ∈ [ T, T ] satisfying this inequality for all j = 1 , . . . , r is equalto T O F Q rk =1 (1 + V k + L√ log log T )(log log T ) α F + + 1 + k V k + L log log T log log T !! × r Y j =1 Z ∞ σ Fj ( X ) − ( V j √ ( n Fj / 2) log log T − L ) e − u / du √ π (5.9)for L√ log log T , k V k ≤ c √ log log T with c sufficiently small.Now, we choose L = 2 rc − k V k + log T and a small enough so that we have the inequalities k V k ≤ a √ log log T and 4 L ≤ (1 + V j ) q n Fj log log T for all j = 1 , . . . , r , where a is the sameconstant as in Proposition 2.1. Then, by equation (5.6) and the estimate R ∞ V e − u / du ≪ V e − V / for V ≥ 0, we obtain Z ∞ σ Fj ( X ) − ( V j √ ( n Fj / 2) log log T − L ) e − u / du √ π ≪ r,F j 11 + V j exp − (cid:18) V j + O r,F j (cid:18) L√ log log T + V j log T log log T (cid:19)(cid:19) ! ≪ F 11 + V j exp − V j O F (cid:18) V j k V k √ log log T + k V k log log T (cid:19)! . Hence, when 0 ≤ V , . . . , V r ≤ a √ log log T with a sufficiently small, (5.9) is ≪ F T r Y j =1 11 + V j + 1(log log T ) α F + exp (cid:18) − V + · · · + V r O F (cid:18) k V k √ log log T (cid:19)(cid:19) . Moreover, we have T exp( − c L ) ≤ T exp( − r k V k ) ≤ T r Y j =1 exp (cid:0) − V + · · · + V r ) (cid:1) ≪ T r Y j =1 11 + V j exp − V j ! . Similarly when θ ∈ [ π , π ] r , except for a set of measure T exp( − c L ) ( L ≫ log T ), the measure of t ∈ [ T, T ] such that Re e − iθ j log F j ( + it ) ≥ V j q n Fj log log T is at least the measure of t ∈ [ T, T ] such thatRe e − iθ j P F j ( + it ) ≥ V j q n Fj log log T + 2 L OINT VALUE DISTRIBUTION OF L -FUNCTIONS 55 When 4 L ≤ V j q n Fj log log T , we have the measure of t satisfying the above inequality for all j =1 , . . . , r is (by Proposition 2.1) T O F Q rk =1 (1 + V k + L√ log log T )(log log T ) α F + + 1 + k V k + L log log T log log T !! × r Y j =1 Z ∞ σ Fj ( X ) − ( V j √ ( n Fj / 2) log log T +2 L ) e − u / du √ π , (5.10)which can be bounded by ≫ F T r Y j =1 11 + V j exp (cid:18) − V + · · · + V r O F (cid:18) k V kL√ log log T (cid:19)(cid:19) when Q rj =1 (1 + V j ) ≤ c (log log T ) α F + with c = c ( F ) > L =2 r k V k + log T and a small enough so that k V k ≤ a √ log log T and 4 L ≤ V j q n Fj log log T holdfor all j = 1 , . . . , r , where a is the same constant as in Proposition 2.1. Then (5.10) is ≫ F T r Y j =1 11 + V j exp (cid:18) − V + · · · + V r O F (cid:18) k V k √ log log T (cid:19)(cid:19) , which completes the proof of Theorem 2.2. (cid:3) We prepare a lemma to prove Theorem 2.4. Lemma 5.2. Let θ ∈ (cid:2) − π , π (cid:3) , and F ∈ S † satisfying (2.1) and (A3). There exist positive constants a = a ( F ) such that for any large V , T meas (cid:8) t ∈ [ T, T ] : Re e − iθ log F ( + it ) > V (cid:9) ≤ exp (cid:18) − a V log log T (cid:19) + exp ( − a V ) . Proof. We can show that, for t ∈ [ T, T ], the inequality Re e − iθ log F (1 / it ) ≤ C log T with C = C ( F ) > X = 3, H = 1 and estimate(3.6). Hence, this lemma holds when V ≥ C log T with C = C ( F ) > 0. In the following, weconsider the case V ≤ C log T . Similarly to the proof of Lemma 5.1, we obtain1 T Z TT (cid:12)(cid:12)(cid:12)(cid:12) log F ( + it ) − P F ( + it, X ) − X | + it − ρ F |≤ X log (( + it − ρ F ) log X ) (cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ A k k k + A k k !(log log T ) k for X = T δ F /k , where δ F has the same meaning as in Lemma 3.11. Moreover, by Lemma 3.6 andLemma 3.11, we obtain 1 T Z TT | P F ( + it, X ) | k dt ≤ ( Ck log log T ) k , and 1 T Z TT X | + it − ρ F |≤ X k dt ≤ C k k k . When V ≤ log log T , we choose k = ⌊ cV / log log T ⌋ , and when V ≥ log log T , we choose k = ⌊ cV ⌋ .Here, c is a suitably small constant depending only on F . Then, by the above calculations and the inequality Re e − iθ log ((1 / it − ρ F ) log X ) ≤ π , we obtain1 T meas (cid:8) t ∈ [ T, T ] : Re e − iθ log F ( + it ) > V (cid:9) ≤ exp (cid:18) − a V log log T (cid:19) + exp ( − a V ) , which completes the proof of Lemma 5.2. (cid:3) Proof of Theorem 2.4. Let 0 ≤ k ≤ a with a = a ( F ) > φ F ( t ) = min ≤ j ≤ r Re e − iθ j log F j (1 / it ) andΦ F ( T, V ) := meas { t ∈ [ T, T ] : φ F ( t ) > V } . Then we have Z TT exp (2 kφ F ( t )) dt = Z ∞−∞ ke kV Φ F ( T, V ) dV. (5.11)We consider the case when θ ∈ [ − π , π ] r first. From Theorem 2.2, it follows that, for any 0 ≤ V ≤ a log log T with a = a ( F ) a suitable constant,Φ F ( T, V ) (5.12) ≪ F T (cid:18) 11 + ( V / √ log log T ) r + 1(log log T ) α F + (cid:19) exp (cid:18) − h F V log log T + C V (log log T ) (cid:19) for some constant C = C ( F ) > F ( T, V ) ≤ T exp (cid:18) − a V log log T (cid:19) + T exp ( − a V ) (5.13)for any large V . Now we choose a = min { a a / , a / } . Put D = 4 a − . We divide the integral onthe right hand side of (5.11) to Z −∞ + Z D k log log T + Z ∞ D k log log T ! ke kV Φ F ( T, V ) dV =: I + I + I , say. We use the trivial bound Φ F ( T, V ) ≤ T to obtain I ≤ T . Also, by inequality (5.13), it followsthat I ≤ T Z ∞ D k log log T k (cid:26) exp (cid:18)(cid:18) − a V log log T + 2 k (cid:19) V (cid:19) + e ( − a +2 k ) V (cid:27) dV ≤ T Z ∞ ke − kV dV ≤ T. Moreover, using inequality (5.12), we find that I ≪ F T Z D k log log T ( E + E ) exp (cid:18) kV − h F V log log T + C V (log log T ) (cid:19) dV ≪ T (log T ) k /h F + C D k Z D k log log T ( E + E ) exp − h F log log T (cid:18) V − kh F log log T (cid:19) ! dV, where E = k V/ √ log log T ) r and E = k (log log T ) α F + 12 . We see that Z D k log log T E exp − h F log log T (cid:18) V − kh F log log T (cid:19) ! dV ≤ k (log log T ) α F + Z ∞−∞ exp (cid:18) − h F log log T V (cid:19) dV ≪ F k (log log T ) α F . OINT VALUE DISTRIBUTION OF L -FUNCTIONS 57 Also, we write Z D k log log T E exp − h F log log T (cid:18) V − kh F log log T (cid:19) ! dV = Z k h F log log T + Z D k log log T k h F log log T ! k exp (cid:18) − h F log log T (cid:16) V − kh F log log T (cid:17) (cid:19) V / √ log log T ) r dV =: I , + I , , say. We find that I , ≪ F k k √ log log T ) r Z ∞−∞ exp (cid:18) − h F log log T V (cid:19) dV ≪ F k √ log log T k √ log log T ) r , and that I , ≤ Z kh F log log T k h F log log T k exp (cid:18) − h F log log T V (cid:19) dV ≤ r log log Th F Z ∞ k √ h F √ log log T ke − u du. If k ≤ (log log T ) − / , the last is clearly ≪ F 1. If k ≥ (log log T ) − / , we apply the estimate R ∞ x e − u du ≪ x − e − x to obtain I , ≤ r log log Th F Z ∞ k √ h F √ log log T ke − u du ≪ . Hence, we obtain I ≪ F T + kT (log T ) k /h F + C D k (cid:18) √ log log T k √ log log T ) r + 1(log log T ) α F + (cid:19) . (5.14)Combing this estimate and the estimates for I , I , we complete the proof of (2.11).Next, we consider the case θ ∈ [ π , π ] r . By equation (5.11), estimate (2.7), and positivity of Φ F ,we have Z TT exp (2 kφ F ( t )) dt ≥ Z ke kV Φ F ( T, V ) dV + Z kh F log log T + √ log log T kh F log log T ke kV Φ F ( T, V ) dV. By estimate (2.7), the first integral on the right hand side is ≫ F T , and the second integral on theright hand side is ≫ F kT k √ log log T ) r Z kh F log log T + √ log log T kh F log log T exp (cid:18) kV − h F V log log T − C V (log log T ) (cid:19) dV ≥ kT (log T ) k h F − C k k √ log log T ) r Z kh F log log T + √ log log T kh F log log T exp − h F log log T (cid:18) V − kh F log log T (cid:19) ! ≫ F kT (log T ) k h F − C k √ log log T k √ log log T ) r , where C ≥ F . Hence, we also obtain Theorem 2.4 in the case θ ∈ [ π , π ] r . (cid:3) Proofs of Theorem 2.3 and Theorem 2.5 Proof of Theorem 2.3. Let F ∈ ( S † ) r and θ ∈ [ − π , π ] r satisfying A . Let T be a sufficiently largeconstant depending on F . Set Y = T K / L where K = K ( F ) > L ≥ (log T ) is a large parameter to be chosen later. Let f be a fixed function satisfying the conditionof this paper (see Notation) and D ( f ) ≥ 2. Assuming the Riemann Hypothesis for F , . . . , F r , weapply Theorem 3.1 with X = Y , H = 1 to obtainlog F j ( + it ) = X ≤ n ≤ Y Λ F j ( n ) v f, ( e log n/ log Y ) n / it log n + X | / it − ρ Fj |≤ Y log(( + it − ρ F j ) log Y ) + R F j ( + it, Y, , where (cid:12)(cid:12) R F j ( + it, Y, (cid:12)(cid:12) ≤ C Y (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Y Λ F j ( n ) w Y ( n ) n + Y + it (cid:12)(cid:12)(cid:12)(cid:12) + d F j log T log Y for any t ∈ [ T, T ]. Here C is a positive constant depending only on f . Moreover, when θ j ∈ (cid:2) − π , π (cid:3) ,it holds that Re e − iθ j X | / it − ρ Fj |≤ Y log(( + it − ρ F j ) log Y ) ≤ π X | / it − ρ Fj |≤ Y , and when θ j ∈ (cid:2) π , π (cid:3) , it holds thatRe e − iθ j X | / it − ρ Fj |≤ Y log(( + it − ρ F j ) log Y ) ≥ − π X | / it − ρ Fj |≤ Y . Hence, there exists some positive constant C > e − iθ j X | / it − ρ Fj |≤ Y log(( + it − ρ F j ) log Y ) ≤ C Y (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Y Λ F j ( n ) w Y ( n ) n + Y + it (cid:12)(cid:12)(cid:12)(cid:12) + d F j log T log Y when θ j ∈ (cid:2) − π , π (cid:3) , and Re e − iθ j X | / it − ρ Fj |≤ Y log(( + it − ρ F j ) log Y ) ≥ − C Y (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Y Λ F j ( n ) w Y ( n ) n + Y + it (cid:12)(cid:12)(cid:12)(cid:12) + d F j log T log Y when θ j ∈ (cid:2) π , π (cid:3) . Taking K = 2( C + C ) max ≤ j ≤ r d F j , we find that there exists some positiveconstant C depending on f such that for any t ∈ [ T, T ] and all j = 1 , . . . , r ,Re e − iθ j log F j ( + it ) ≤ Re e − iθ j X ≤ n ≤ Y Λ F j ( n ) v f, ( e log n/ log Y ) n / it log n (6.1)+ C log Y (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Y Λ F j ( n ) w Y ( n ) n + Y + it (cid:12)(cid:12)(cid:12)(cid:12) + L OINT VALUE DISTRIBUTION OF L -FUNCTIONS 59 when θ j ∈ (cid:2) − π , π (cid:3) , andRe e − iθ j log F j ( + it ) ≥ Re e − iθ j X ≤ n ≤ Y Λ F j ( n ) v f, ( e log n/ log Y ) n / it log n − C log Y (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Y Λ F j ( n ) w Y ( n ) n + Y + it (cid:12)(cid:12)(cid:12)(cid:12) − L θ j ∈ (cid:2) π , π (cid:3) .Put X = Y / (log log T ) r +1) . By Lemma 3.6 and assumption (A1), we obtain Z TT (cid:12)(cid:12)(cid:12)(cid:12) X X 0. Similarly to the proofs of estimates (3.23), we can show thatfor any integer k with 1 ≤ k ≤ L / K Z TT (cid:12)(cid:12)(cid:12)(cid:12) X X 0. Moreover, by Lemma 3.10, we have Z TT C log Y (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Y Λ F j ( n ) w Y ( n ) n + Y + it (cid:12)(cid:12)(cid:12)(cid:12) k dt ≤ T C k k k for any integer k with 1 ≤ k ≤ L / K and for some constant C = C ( F j ) > 0. Here the assumptionsin Lemma 3.10 is satisfied as we can take κ F arbitrarily large. Therefore, the set of t ∈ [ T, T ] suchthat for all j = 1 , . . . , r , L ≤ (cid:12)(cid:12)(cid:12)(cid:12) X X 0, estimate (4.22), and Proposition 2.2, we find that if θ j ∈ [ − π , π ],1 T meas( S ( T, V ; F , θ )) ≪ F T meas( X )+ (cid:18) V · · · V r + 1(log log T ) α F + (cid:19) r Y j =1 exp − V j − V j σ F j ( X ) σ F j V j σ F j ( X ) ! ++ O F k V kL√ log log T + L log log T + (cid:18) k V k√ log log T (cid:19) − ϑ F − ϑ F ! ≪ F (cid:18) V · · · V r + 1(log log T ) α F + (cid:19) r Y j =1 exp − V j O F (cid:18) k V k √ log log T log k V k (cid:19)! for k V k ≤ a V / m (log log T ) / (log T ) / . Hence, we obtain estimate (2.8). Similarly, we can alsofind that if θ j ∈ [ π , π ],1 T meas( S ( T, V ; F , θ )) ≫ F (cid:18) V · · · V r + 1(log log T ) α F + (cid:19) r Y j =1 exp − V j − V j σ F j ( X ) σ F j V j σ F j ( X ) ! + − O F k V kL√ log log T + L log log T + (cid:18) k V k√ log log T (cid:19) − ϑ F − ϑ F ! − T meas( X ) ≫ F V · · · V r exp (cid:18) − V + · · · + V r − O F (cid:18) k V k √ log log T log k V k (cid:19)(cid:19) for k V k ≤ a V / m (log log T ) / (log T ) / satisfying Q rj =1 V j ≤ a (log log T ) α F + with a = a ( F ) > θ j ∈ (cid:2) − π , π (cid:3) . Putting L = K log T log log T , we see that Y = (log T ) / ,and hence there exists a positive constant A = A ( F ) such that the right hand side of (6.1) is ≤ A log T √ log log T q n Fj log log T uniformly for any t ∈ [ T, T ] and all j = 1 , . . . , r . Thus, we may assume OINT VALUE DISTRIBUTION OF L -FUNCTIONS 61 k V k ≤ A log T √ log log T . We first consider the case when √ log log T ≤ k V k ≤ A log T √ log log T . Set L = b k V k √ log log T , where b is some small positive constant such that the inequality Y ≥ X ) ≪ F T exp (cid:16) − c k V k p log log T log k V k (cid:17) for some constant c = c ( F ) > 0. Using Lemma 3.6, we have, uniformly for any j = 1 , . . . , r , Z TT | P F j ( + it, X ) | k dt ≪ F T ( C k log log T ) k (6.4)for any integer k with 1 ≤ k ≤ L log log T and some C = C ( F ). Combing (6.3) and (6.4), we obtain1 T meas( S ( T, V ; F , θ )) ≪ min ≤ j ≤ r T meas (cid:8) t ∈ [ T, T ] : Re e − iθ j log F j ( + it ) > V j (cid:9) ≪ F k V k − k C k k k + exp (cid:16) − c k V k p log log T log k V k (cid:17) . When k V k ≤ log log T , we choose k = ⌊ c k V k ⌋ , and when k V k > log log T , we choose k = ⌊ c k V k√ log log T ⌋ , where c is a suitably small positive constant depending only on F . Then, itfollows that1 T meas( S ( T, V ; F , θ )) ≪ F exp (cid:0) − c k V k (cid:1) + exp (cid:16) − c k V k p log log T log k V k (cid:17) , which completes the proof of (2.10). (cid:3) Proof of Theorem 2.5. Let T be large, and put ε ( T ) = (log T ) − . Let k ≥ 0. We recall equation(5.11), which is Z TT exp (2 kφ F ( t )) dt = Z ∞−∞ ke kV Φ F ( T, V ) dV. We divide the integral on the right hand side to Z −∞ + Z D k log log T + Z ∞ D k log log T ! ke kV Φ F ( T, V ) dV =: I + I + I , say. Here, D = D ( F ) is a suitably large positive constant. Now we consider the case when θ j ∈ [ − π , π ]. We use the trivial bound Φ F ( T, V ) ≤ T to obtain I ≤ T . Applying estimate (2.10), wefind that the estimate Φ F ( T, V ) ≪ F T exp ( − kV )holds for V ≥ D k log log T when D is suitably large. Therefore, we have I ≪ F T Z ∞ D log log T ke − kV dV ≪ T. By estimate (2.8), we find thatΦ F ( T, V ) ≪ k, F T (cid:18) 11 + ( V / √ log log T ) r + 1(log log T ) α F + (cid:19) exp (cid:18) − h F V log log T + C V (log log T ) log T (cid:19) ≪ T (cid:18) 11 + ( V / √ log log T ) r + 1(log log T ) α F + (cid:19) (log T ) C D k ε ( T ) exp (cid:18) − h F V log log T (cid:19) for (log log T ) / ≤ V ≤ D k log log T . Here, C = C ( F ) is some positive constant. Similarly to theproof of (5.14) by using this estimate, we obtain I ≪ F T + T (log T ) k /h F + Bk ε ( T ) (cid:18) k √ log log T k √ log log T ) r + 1(log log T ) α F + (cid:19) . Hence, we obtain (2.13).For estimate (2.14), it holds from the positivity of Φ F ( T, V ) and equation (5.11) that Z TT exp (2 kφ F ( t )) dt ≫ k Z kh F log log T + √ log log T kh F log log T e kV Φ F ( T, V ) dV. When θ i ∈ [ π , π ], assuming ϑ F < r +1 , we use (2.9) to obtainΦ F ( T, V ) ≫ k, F T V / √ log log T ) r exp (cid:18) − h F V log log T − CV (log log T ) log T (cid:19) ≫ F T (log T ) − C k ε ( T ) V / √ log log T ) r exp (cid:18) − h F V log log T (cid:19) for kh F log log T ≤ V ≤ kh F log log T + √ log log T . Here, C = C ( F ) is a positive constant. Similarlyto the proof of (2.12) by using this estimate and the bound Φ F ( T, V ) ≫ F T for 0 ≤ V ≤ 1, we canalso obtain (2.14). (cid:3) Concluding remarks Moments of max/min values. Under the same conditions as in Theorem 2.4, we can applyTheorem 2.2 to show that for sufficiently small k , there exists some constant B = B ( F ) > Z TT (cid:18) max ≤ j ≤ r | F j ( + it ) | (cid:19) k dt ≪ F T (log T ) n F k + Bk , (7.1) Z TT (cid:18) min ≤ j ≤ r | F j ( + it ) | (cid:19) − k dt ≫ F T (log T ) n F k − Bk , where n F = max ≤ j ≤ r n F j . Assuming the Riemann Hypothesis for these L -functions, we can replacethe term Bk in the exponent by any ε > k > 0. In fact, consider S j = (cid:8) t ∈ [ T, T ] : Re e − iθ j log F j ( + it ) > V (cid:9) . Using the inclusion-exclusion principle, we see that r X j =1 meas S j − X ≤ ℓ 2) log log T for j = 1 , . . . , r , we obtain1 T meas (cid:26) t ∈ [ T, T ] : max ≤ j ≤ r Re e − iθ j log F j ( + it ) > V (cid:27) ≪ F 11 + V / √ log log T exp (cid:18) − V n F log log T (cid:18) O F (cid:18) V log log T (cid:19)(cid:19)(cid:19) OINT VALUE DISTRIBUTION OF L -FUNCTIONS 63 for any θ ∈ [ − π , π ] r and 0 ≤ V ≤ a log log T with a = a ( F ) > r = 1 , V = V √ ( n Fj / 2) log log T for j = 1 , . . . , r , and( V , V ) = (cid:18) V √ ( n Fℓ / 2) log log T , V √ ( n Fm / 2) log log T (cid:19) for 1 ≤ ℓ < m ≤ r , we obtain1 T meas (cid:26) t ∈ [ T, T ] : max ≤ j ≤ r Re e − iθ j log F j ( + it ) > V (cid:27) ≫ F 11 + V / √ log log T exp (cid:18) − V n F log log T (cid:18) O F (cid:18) V log log T (cid:19)(cid:19)(cid:19) for any θ ∈ [ π , π ] r and 0 ≤ V ≤ a min { log log T , (log log T ) α F r + + r } . The rest of the argumentfollows the same way as in Theorems 2.4 and Theorem 2.5. Note that assumptions (2.2), (A2) are notneeded for (7.1).7.2. Moments of product of L -functions. Our method can be modified to study the moment ofproduct of L -functions. Under the same assumptions as in Theorem 2.4, we can also show that forsufficiently small k , . . . , k r > B = B ( F ) such that Z TT r Y j =1 | F j ( + it ) | k j dt ≪ F T (log T ) n F k + ··· + n Fr k r + BK , (7.3) Z TT r Y j =1 | F j ( + it ) | − k j dt ≫ F T (log T ) n F k + ··· + n Fr k r − BK , if θ F ≤ r + 1 (7.4)where K = max ≤ j ≤ r k j . If we assume the Riemann Hypothesis for these L -functions, the term BK in the exponent can be replaced by any ε > k , . . . , k r > S X ( T, V ; F , θ ). Let X ≥ k = ( k , . . . , k r ) ∈ ( R ≥ ) r , and let F be a r -tuple of Dirichlet series and θ ∈ R r satisfying (S4), (S5), (A1), and (A2).Let σ F ( X, k ) = vuut r X j =1 k j X p ≤ X ∞ X ℓ =1 | b F j ( p ℓ ) | p ℓ and S X ( T, V ; F , θ , k ) = 1 T meas ( t ∈ [ T, T ] : P rj =1 k j Re e − iθ j P F j ( + it, X ) σ F ( X ; k ) > V ) . Then, we can show that for any k ∈ ( R > ) r and | V | ≤ aσ F ( X ; k ) with a = a ( F , k ) > T meas ( t ∈ [ T, T ] : P rj =1 k j Re e − iθ j P F j ( + it, X ) σ F ( X ; k ) > V ) ∼ Z ∞ V e − u / du √ π by modifying the proofs in Section 4.3. Based on this asymptotic formula, we can also prove ananalogue of Theorem 2.2 and thus (7.3). Similarly, we can also obtain (7.4).7.3. Other large deviation results. We would like to mention that our method also recovers thework of Heuberger-Kropf [16] for higher dimensional quasi-power theorem, and it is likely our methodyields improvement to their work in the direction of large deviations. References [1] M. 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Zhang, Integral mean values of Maass L -functions, Int. Math. Res. Not. , Art. ID 41417, 19 pp.(S. Inoue) Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602,Japan Email address : [email protected] (J. Li) Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Email address ::