Explicit Upper Bounds for L-functions on the critical line
aa r X i v : . [ m a t h . N T ] J un EXPLICIT UPPER BOUNDS FOR L-FUNCTIONS ON THECRITICAL LINE
VORRAPAN CHANDEE
Abstract.
We find an explicit upper bound for general L -functions on thecritical line, assuming the Generalized Riemann Hypothesis, and give as illus-trative examples its application to some families of L -functions and Dedekindzeta functions. Further, this upper bound is used to obtain lower bounds be-yond which all eligible integers are represented by Ramanujan’s ternary formand Kaplansky’s ternary forms. This improves on previous work of Ono andSoundararajan [7] on Ramanujan’s form and Reinke [8] on Kaplansky’s formwith a substantially easier proof. Introduction
Finding upper bounds for L -functions on the critical line is an interesting prob-lem in analytic number theory. One classical bound is the convexity bound, whichfollows from the Phragmen-Lindel¨of principle and the approximate functional equa-tion of the L-function. For any ǫ > , the convexity bound is L ( f, s ) ≪ q ( f, s ) / ǫ , where ℜ s = and q ( f, s ) is the analytic conductor of the L -function, whose defi-nition we recall later. However, for many applications this convexity bound is notsufficient, and one needs to improve it for relevant L -functions by reducing theexponent 1 / ǫ > ,L ( f, s ) ≪ q ( f, s ) ǫ , where ℜ s = , and the implied constant depends on ǫ. This is a well known conse-quence of the Generalized Riemann Hypothesis (GRH) (see e.g. Corollary 5.20 p.116 in [5]). Indeed, in the process of deducing the Lindel¨of Hypothesis from GRH,we obtain that for some
A > L ( f, ) ≪ exp (cid:18) A log q ( f, )log log q ( f, ) (cid:19) . In this paper, instead of giving asymptotic upper bounds for L -functions, we willcompute an explicit upper bound for L -functions on the critical line, assumingGRH. This type of upper bound is useful for certain applications. For example,Ono and Soundararajan [7] used explicit upper bounds (on GRH) for L -functionsat the critical point to show that all odd numbers larger than 2 · are represented Mathematics Subject Classification.
Primary 11M41; Secondary 11E25.
Key words and phrases.
L-functions, critical line, ternary quadratic form. by Ramanujan’s form x + y + 10 z . Using our work here we can give a substan-tially easier proof of their results. Moreover our bounds show that all odd numberslarger than 3 · are represented, so that much less computation is required. Weanticipate that these results would be useful in studying representation questionsfor general ternary quadratic froms. We will give more details on this problem,together with other examples in § | L ( f, ) | as a short sumover prime powers and a sum over non-trivial zeros of L ( f, s ). Furthermore in find-ing our upper bound for log | L ( f, ) | , the contribution of nontrivial zeros can beignored. The short sum over prime powers can be explicitly bounded both numer-ically and in term of analytic conductor, which will be defined below. We will givedetails of the proof in §
2. Before we state the main theorem and corollaries, hereare basic notations that we will use throughout the paper.From [5] p. 94, L ( f, s ) is an L -function if it satisfies the following properties.I. It is a Dirichlet series with Euler product of degree d ≥ , (2) L ( f, s ) = X n ≥ λ f ( n ) n − s = Y p (1 − α ( p ) p − s ) − ... (1 − α d ( p ) p − s ) − , and(3) − L ′ L ( f, s ) = X n = p l l ≥ ( α l ( p ) + ... + α ld ( p )) log pn s = X n a ( n ) n s with λ f (1) = 1 , | α p | < p for all p . The series and Euler products are absolutelyconvergent for ℜ ( s ) > . II. Let(4) γ ( f, s ) = π − ds/ d Y j =1 Γ (cid:18) s + k j (cid:19) be a gamma factor. Since we will assume GRH, ℜ k j ≥ . This condition tells usthat γ ( f, s ) has no zero in C and no pole for ℜ ( s ) ≥ . Let an integer q ( f ) ≥ L ( f, s ) such that α i ( p ) = 0 for p not dividing q ( f ) . LetΛ( f, s ) = q ( f ) s/ γ ( f, s ) L ( f, s ) L ( f, s ) must satisfy the functional equation(5) Λ( f, s ) = ǫ ( f )Λ( f, − s ) , where | ǫ ( f ) | = 1, and Λ( f, s ) = Λ( f, s ). We define the analytic conductor of L ( s, f )to be(6) C = q ( f ) π d d Y j =1 (cid:12)(cid:12)(cid:12)(cid:12)
14 + k j (cid:12)(cid:12)(cid:12)(cid:12) . In this paper, we will assume that Λ( f, s ) has no zero or pole at ℜ s = 1 . Note thatby the functional equation, this implies the same holds at at ℜ s = 0 . XPLICIT UPPER BOUNDS FOR L-FUNCTIONS ON THE CRITICAL LINE 3
Our main theorem will be an upper bound for log L ( f, ) in terms of the sumover prime powers and the analytic conductor (Theorem 2.1 in § A in (1) as below. Corollary 1.1.
Let L ( f, s ) be an L -function satisfying the conditions above and C be defined as in (6). Furthermore assume that L ( f, s ) satisfies Ramanujan’sconjecture. Then for log log C ≥ , log | L ( f, ) | ≤ d
25 log C log log C + 38 log C log log C ,
Remark 1.
Without Ramanujan’s conjecture, we instead obtain log | L ( f, ) | ≤
98 log C log log C + O (cid:0) d log C log log C (cid:1) . Remark 2.
It may be possible to improve the explicit constant / appearing inCorollary 1.1 by using a different kernel in the proof of Lemma 2.4. Applying Corollary 1.1 to families of L -functions which satisfy Ramanujan’sconjecture, we then easily obtain the explicit upper bound for these L -functions interm of their analytic conductor. We will illustrate our results using three examplesof families of L -functions: Dirichlet L -functions, holomorphic cusp forms and theRiemann zeta function with varying height on the critical line. Corollary 1.2.
Let χ be a primitive even Dirichlet character modulo q . Let ˜ C = q (cid:0) | t | + (cid:1) . If log log ˜ C ≥ , then for any t, | L ( + it, χ ) | ≤ exp C log log ˜ C + 38 log ˜ C log log ˜ C ! . Corollary 1.3.
Let f ( s ) = P n ≥ λ ( n ) q n be a holomorphic cusp form, of weight k ≥ and level q , and L ( f, s ) = P n ≥ λ ( n ) n s be an L -function associated to f. Define ˜ C = qk . If log log ˜ C ≥ , then | L ( f, ) | ≤ exp C log log ˜ C + 38 log ˜ C log log ˜ C ! . Corollary 1.4.
Assume T ≤ t ≤ T. Let ˜ C = T . Also assume that log log ˜ C ≥ .Then | ζ ( + it ) | ≤ exp C log log ˜ C + 38 log ˜ C log log ˜ C ! . Remark 3.
The analytic conductor is a measure of the complexity of the L -function. For our examples above, C ≈ ˜ C. Hence we can apply Corollary 1.1to ˜ C. The details will be discussed in § Finally with some modifications, we can apply Corollary 2.1 to Dedekind zetafunctions. We will prove the following Corollary in § Corollary 1.5.
Let K/ Q be a number field of degree d . The Dedekind zeta function ζ K ( s ) = Y p (1 − ( N p ) − s ) − is an L -function of degree d with conductor q equal to the absolute value of thediscriminant of K . If log log C ≥ then | ζ K ( ) | ≤ .
33 exp (cid:18) d
25 log C log log C + 38 log C log log C (cid:19) . Acknowledgements: I am very grateful to Professor Soundararajan for his guid-ance throughout the making of this paper. I would like to thank Xiannan Li forhelpful editorial comments. I also want to thank the referee for careful reading ofthis paper, and helpful comments.2.
Proof of the Main Theorem and its Corollaries
We will now prove an explicit upper bound for L ( f, ) in terms of a sum overpowers of primes and parameters associated with L ( f, s ). Then we will simplify thesum and those constants to derive the corollaries stated in the introduction. Ourmain theorem is Theorem 2.1.
Assume GRH and Λ( f, s ) has no pole or zero at s = 0 , . Let λ = 0 . ... denote the unique positive real number satisfying e − λ = λ + λ / . Then for all log x ≥ λ ≥ λ , and log x ≥ we have | L ( f, ) | ≤ ℜ X n ≤ x a ( n ) n
12 + λ log x log n log xn log x + (cid:18) λ (cid:19) log C log x + ( λ + λ ) d log x + 4 de − λ x / log x . Remark 4.
The above easily leads to an upper bound for log (cid:12)(cid:12) L (cid:0) f, + it (cid:1)(cid:12)(cid:12) as well.Indeed, we can set L new ( f, s ) = L ( f, s + it ) , so that L (cid:0) f, + it (cid:1) = L new (cid:0) f, (cid:1) .L new ( f, s ) satisfies the functional equation (5), but with k j + it in the gamma factorin (4). Proof of Theorem 2.1
Let ρ = + iγ run over the non-trivial zeros of L ( f, s ) . Define G ( s ) = ℜ X ρ s − ρ = X ρ σ − ( σ − ) + ( γ − t ) . To prove Theorem 2.1, we need to use the Hadamard factorization formula below.The proof can be found in [5] (see Theorem 5.6).
Proposition 2.2.
Let L ( f, s ) be an L -function, and s = σ + it . There exist con-stants a, b such that Λ( f, s ) = e a + bs Y ρ =0 , (cid:18) − sρ (cid:19) e s/ρ . where ρ runs over all zeros of Λ( f, s ) . Hence (7) − L ′ L ( f, s ) = 12 log qπ d + 12 d X j =1 Γ ′ Γ (cid:18) s + k j (cid:19) − b − X ρ =0 , (cid:18) s − ρ + 1 ρ (cid:19) is uniformly and absolutely convergent in compact subsets which have no zeros orpoles. Furthermore ℜ ( − b + P ρ ) = 0 . XPLICIT UPPER BOUNDS FOR L-FUNCTIONS ON THE CRITICAL LINE 5
By (7), since ℜ ( − b + P ρ ) = 0 , if L ( f, s ) = 0 we have(8) − ℜ L ′ L ( f, s ) = 12 log qπ d + 12 d X j =1 ℜ Γ ′ Γ (cid:18) s + k j (cid:19) − G ( s ) . We need the following lemma to find upper bound for ℜ Γ ′ Γ (cid:16) s + k j (cid:17) . The proof ofthe lemma can be found in Appendix.
Lemma 2.3.
Let z = x + iy, where x ≥ . Then ℜ Γ ′ Γ ( z ) ≤ log | z | . From Lemma 2.3 and (8), we obtain(9) − ℜ L ′ L ( f, s ) ≤
12 log qπ d + 12 d X j =1 log (cid:12)(cid:12)(cid:12)(cid:12) s + k j (cid:12)(cid:12)(cid:12)(cid:12) − G ( s ) . When we integrate the inequality above as σ varies from to σ = + λ log x , weget log | L ( f, ) | − log | L ( f, σ ) | (10) ≤ λ x log qπ d + 12 d X j =1 Z σ log (cid:12)(cid:12)(cid:12)(cid:12) σ + k j (cid:12)(cid:12)(cid:12)(cid:12) dσ − X ρ log ( σ − ) + γ γ ≤ λ x log qπ d + (cid:18) λ x (cid:19) d X j =1 log (cid:12)(cid:12)(cid:12)(cid:12) σ + k j (cid:12)(cid:12)(cid:12)(cid:12) − ( σ − ) G ( σ ) ≤ λ x log C + λ d log x − λ x G ( σ ) , where we use the fact that log(1 + x ) ≥ x x and for λ ≤ log x , (11)log (cid:12)(cid:12)(cid:12)(cid:12) σ + k j (cid:12)(cid:12)(cid:12)(cid:12) ≤ log (cid:12)(cid:12)(cid:12)(cid:12)
14 + k j (cid:12)(cid:12)(cid:12)(cid:12) + log (cid:12)(cid:12)(cid:12)(cid:12) λ (log x )( + k j ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ log (cid:12)(cid:12)(cid:12)(cid:12)
14 + k j (cid:12)(cid:12)(cid:12)(cid:12) + 2 λ log x . To prove Theorem 2.1, we need an upper bound for log | L ( f, σ ) | , which we willobtain from Lemma 2.4 below. This lemma is a version of Lemma 1 in [10] forgeneral L -functions. The proof of Lemma 2.4 is essentially the same as the one in[10]; however we provide the sketch of the proof here for completeness. Lemma 2.4.
Unconditionally, for any s not conciding with 1, 0 or a zero of L ( f, s ) ,and for any x ≥ , we have − L ′ L ( f, s ) = X n ≤ x a ( n ) n s log xn log x + 1log x (cid:18) L ′ L ( f, s ) (cid:19) ′ + 1log x X ρ =0 , x ρ − s ( ρ − s ) + 1log x d X j =1 ∞ X n =0 x − n − k j − s (2 n + k j + s ) , where a ( n ) is defined in (3). VORRAPAN CHANDEE
Proof.
With c = max(1 , − σ ) , integrating term by term using the Dirichlet seriesexpansion of − L ′ L ( f, s ), we have12 πi Z c + i ∞ c − i ∞ − L ′ L ( f, s + w ) x w w dw = X n ≤ x a ( n ) n s log xn . On the other hand, moving the line of integration to the left and calculating residuesgiveslog x − L ′ L ( f, s ) − x (cid:18) L ′ L ( f, s ) (cid:19) ′ − x X ρ =0 , x ρ − s ( ρ − s ) − x d X j =1 ∞ X n =0 x − n − k j − s (2 n + k j + s ) . Equating these two expressions, we obtain the lemma. (cid:3)
We take s = σ in Lemma (2.4), extract the real parts of both sides, and integrateover σ from σ to ∞ . Thus for x ≥ | L ( f, σ ) | = ℜ (cid:0) X n ≤ x a ( n ) n σ log n log xn log x − x L ′ L ( f, σ ) + 1log x X ρ =0 , Z ∞ σ x ρ − σ ( ρ − σ ) dσ + 1log x Z ∞ σ d X j =1 ∞ X n =0 x − n − k j − σ (2 n + k j + σ ) dσ (cid:1) . Moreover X ρ (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ σ x ρ − σ ( ρ − σ ) dσ (cid:12)(cid:12)(cid:12)(cid:12) ≤ X ρ Z ∞ σ x − σ | ρ − σ | dσ = X ρ x − σ | ρ − σ | log x = e − λ G ( σ ) λ , and for ℜ k j ≥ , ℜ Z ∞ σ d X j =1 ∞ X n =0 x − n − k j − σ (2 n + k j + σ ) dσ ≤ d X j =1 x −ℜ k j − σ log x | k j + σ | ∞ X n =0 x − n ≤ de − λ x / log x . Hence using the previous two lines and (9), we deduce thatlog | L ( f, σ ) | (12) ≤ ℜ X n ≤ x a ( n ) n σ log n log xn log x + 1log x
12 log qπ d + 12 d X j =1 log (cid:12)(cid:12)(cid:12)(cid:12) σ + k j (cid:12)(cid:12)(cid:12)(cid:12) − G ( σ ) + 4 de − λ x / log x + e − λ G ( σ ) λ log x . Adding the inequalities (10), (12) and using (11) we getlog | L ( f, ) | ≤ ℜ X n ≤ x a ( n ) n σ log n log xn log x + (cid:18) λ + 12 log x (cid:19) log C + ( λ + λ ) d log x (13) + 4 de − λ x / log x + G ( σ )log x (cid:18) e − λ λ − − λ (cid:19) . For λ ≥ λ , the term involving G ( σ ) above gives a negative contribution, and wecan omit it. Hence the theorem is proved. Proof of Corollary 1.1:
XPLICIT UPPER BOUNDS FOR L-FUNCTIONS ON THE CRITICAL LINE 7
Since our L -function L ( f, s ) satisfies Ramanujan’s conjecture, | a ( n ) | ≤ d Λ( n ) . FromTheorem 2.1 picking λ = 0 .
5, we have(14)log | L ( f, ) | ≤ X n ≤ x d Λ( n ) n
12 + 12 log x log n log xn log x + 34 log x log C + d (cid:18)
34 log x + 4 e − . x / log x (cid:19) . When log x ≥ ,
34 log x + e − . x / log x ≤ . . Now we need to find the upper boundof the sum over powers of primes. We shall prove the following inequality(15) X n ≤ x Λ( n ) n
12 + λ log x log n log xn log x ≤ . x / log x . Let f ( t ) = t + λ log x log t log xt log x . By partial summation, X n ≤ x Λ( n ) n
12 + λ log x log n log xn log x = − Z x ( X n ≤ t Λ( n )) f ′ ( t ) dt Since we assume GRH, using the result of Lemma 8 and (3.21) in [9] (pick δ = √ √ x ), we obtain that for t > , X n ≤ t Λ( n ) ≤ t + 0 . p √ t + 2 √ t ! t For t ≥ , we have X n ≤ t Λ( n ) < (1 . t. The inequality is also true for 10 > t ≥ X n ≤ x Λ( n ) n
12 + λ log x log n log xn log x ≤ − . Z x tf ′ ( t ) dt = 1 . (cid:18) x / / (2 log x ) log 2 log x + Z x f ( t ) dt (cid:19) We change variable t = x/y and obtain Z x f ( t ) dt = e − λ x / log x Z x/ log yy / − / (2 log x ) (1 − log y log x ) dy ≤ . e − λ x / log x , where the last inequality follows because the second integral above is a decreasingfunction of x when x > . The constant appearing on the right hand side isderived by substituing x = e in the integral. Finally by absorbing the constantterm into x / log x term, we derive (15). Choosing x = log C in (14) and applying(15), we prove the corollary. Proof of Corollary 1.2:
The Dirichlet L -function L ( s + it, χ ) is an L -functionof degree 1 with conductor q and satisfies Ramanujan’s conjecture. The gammafactor is γ ( s ) = π − s/ Γ (cid:0) s + it (cid:1) , so the analytic conductor of L ( s + it, χ ) as definedin (6) is C = q π q + t . For any t , log C ≤ log q (cid:0) + | t | (cid:1) . Therefore Corollary 1.1
VORRAPAN CHANDEE can be applied to ˜ C = q (cid:0) + | t | (cid:1) , and we then obtain the corollary. Proof of Corollary 1.3: L ( f, s ) is an L -function of degree 2 with conductorq and gamma factor γ ( f, s ) = π − s Γ (cid:18) s + ( k − / (cid:19) Γ (cid:18) s + ( k + 1) / (cid:19) .L ( f, s ) satisfies Ramanujan’s conjecture and its analytic conductor C is qπ (cid:0) k − (cid:1) (cid:0) k + (cid:1) . For k ≥
1, we havelog C = log qk − log 4 π + log (cid:0) k − k (cid:1) ≤ log ˜ C − log 4 π + k − k ≤ log ˜ C. Therefore we can apply Corollary 1.1 to ˜ C = qk and get the result. Proof of Corollary 1.4
The Riemann zeta function ζ ( s + it ) is an L -functionof degree 1. The gamma factor is γ ( s ) = π − s/ Γ (cid:0) s + it (cid:1) . We can still apply Corol-lary 1.1, but we need to add a constant term derived from the pole s = 1 − it . Theanalytic conductor C is π q + t . For T ≤ t ≤ T and e ≤ (log ˜ C ) ≤ T , log ˜ C ≤ log t + log 12 π + 18 t ≤ log T + log 1 π + 18 t ≤ log T. Therefore we can apply Corollary 1.1 to ˜ C = T. Now we consider (7). For ζ ( s + it )it becomes − ζ ′ ζ ( s + it ) = 12 log qπ d + 12 d X j =1 Γ ′ Γ (cid:18) s + k j (cid:19) − b − X ρ =0 , (cid:18) s − ρ + 1 ρ (cid:19) + 1 s − it + 1 s + it . Following the proof of Theorem 2.1, we obtain that the contribution of the poleterms is bounded by 1 . · − , which is negligible when we apply Corollary 1.1.3. Dedekind Zeta Functions
Our upper bound can also be applied to Dedekind zeta functions. In this section,we will prove Corollary 1.5.
Proof.
The gamma factor of ζ K ( s ) is γ ( s ) = π − ds/ Γ (cid:16) s (cid:17) r + r Γ (cid:18) s + 12 (cid:19) r where r is the number of real embeddings of K and r the number of pairs ofcomplex embeddings, so that d = r + 2 r . Let H ( s ) = ζ K ( s )( s − . Since the Dedekind zeta function has a simple pole at s = 1 , H ( s ) is entire. The equation (8) for H ( s ) is −ℜ H ′ H ( s ) = 12 log qπ d + r + r ℜ Γ ′ Γ (cid:16) s (cid:17) + r ℜ Γ ′ Γ (cid:18) s + 12 (cid:19) + ℜ s − G ( s ) . Similar to (10), we integrate the inequality above from as σ varies from to σ ,use Lemma 2.3, and obtain(16) log | H ( ) | − log | H ( σ ) | ≤ log C x + d x − ( σ − ) G ( σ ) + 1log x . Now we need an upper bound for log | H ( σ ) | , which will be derived in the sameway as the upper bound for log | L ( f, σ ) | . XPLICIT UPPER BOUNDS FOR L-FUNCTIONS ON THE CRITICAL LINE 9
Lemma 3.1.
Unconditionally, for any positive real number t > and any x ≥ we have − H ′ H ( t ) ≤ X n ≤ x d Λ( n ) n t log xn log x + 1log x (cid:18) H ′ H ( t ) (cid:19) ′ + 1log x X ρ =0 , x ρ − t ( ρ − t ) + ( r + r )log x ∞ X n =0 x − n − t (2 n + t ) + r log x ∞ X n =0 x − n − − t (2 n + 1 + t ) . Proof.
By definition of H ( s ) and by using the same arguments as in Lemma 2.4,we have − H ′ H ( t ) = − ζ ′ K ζ K ( t ) − t − ≤ X N a ≤ x Λ K ( a )( N a ) t log xN a log x + 1log x (cid:18) H ′ H ( t ) (cid:19) ′ + 1( t − log x + 1log x X ρ =0 , x ρ − t ( ρ − t ) + ( r + r )log x ∞ X n =0 x − n − t (2 n + t ) + r log x ∞ X n =0 x − n − − t (2 n + 1 + t ) − x − t t log x − x − t ( t − log x − t − . Furthermore X N a ≤ x Λ K ( a )( N a ) t log xN a log x ≤ X n ≤ x d Λ( n ) n t log xn log x , and s − log x − x − s ( s − log x − s − and − x − s s log x are entire when ℜ s ≥ t and x ≥ . Combining thesefacts and the inequalities above, we prove the lemma. (cid:3)
We take t = σ in Lemma 3.1 and integrate over t from σ to ∞ . By the samereasoning as in the proof of Theorem 2.1,log | H ( ) | ≤ X n ≤ x d Λ( n ) n
12 + 12 log x log n log xn log x + 3 log C x + 3 d x + 4 de − . x / log x + 2log x + 1 + 1log x . From Corollary 1.1 (let x = log C ), we obtainlog | ζ K ( ) | ≤ d
25 log C log log C + 38 log C log log C + 32 log log C + log 2 . The lemma follows because log log C ≥ . (cid:3) Application to Positive Definite Ternary Quadratic Forms
We call N eligible for a positive ternary quadratic form f ( x, y, z ) if there are nocongruence obstructions prohibiting f from representing N . Let χ = (cid:0) − N · (cid:1) bethe Kronecker-Legendre symbol. Also define L ( s, χ ) := ∞ X n =1 χ ( n ) n s , and L ( E ( − N ) , s ) := ∞ X n =1 A ( n ) χ ( n ) n s . By Dirichlet’s class number formula (see [2]) and special case of Waldspurger’stheorem connecting the Fourier coefficients of half-integral weight cusp forms and L ( E ( − N ) ,
1) (Theorem 2 of [7]), Ono and Soundararajan showed that if N is an eligible square-free integer coprime to 10 and is not represented by Ramanujan’sform, then(17) L ( E ( − N ) , L (1 , χ ) ≥ (cid:16) q π (cid:17) / , where q is a conductor of E ( − N ) , and its value is 1600 N (see Prop.2 in [7]).Ono and Soundarajan proved that (17) failed when N ≥ · . Note that for N ≤ · , W.Galway verified by computer whether it is represented by the form.The difficulty of showing this is finding an upper bound for L ( E ( − N ) ,
1) becauseof the big contribution of zeros on the critical line. It involves long and complicatedcalculations.As seen in the proof of Theorem 2.1, the technique from [10] allows us to ignore thecontribution of nontrivial zeros. Therefore by applying upper bound in Theorem2.1, not only is our calcuation much simpler but also we get a better lower boundfor N , i.e. it requires N ≥ · to yield contradiction for (17).With Ono and Soundararajan’s methods of deriving (17) and the upper boundin Theorem 2.1, under GRH, we may be able to obtain not too large positive in-teger N such that if m ≥ N and is represented by the spinor genus of a positivedefinite ternary form, then it is represented by the form itself. For example, inthis paper, we will apply those techniques to Kaplansky’s form and show that if N ≥ · is a squarefree integer, then it is represented by Kaplansky’s forms: ϕ ( x, y, z ) = x + y + 7 z and ϕ ( x, y, z ) = x + 2 y + 2 yz + 4 z . This resultwill have a simpler proof and a better lower bound than the one that T.Reinke [8]derived.T. Reinke [8] applied the method in [7] to prove the analog of (17). If N is aneligible square-free integer and is not represented by ϕ j , where j = 1 , , then(18) L ( E ( − N ) , L (1 , χ d ) ≥ (cid:26) q N π if ( N,
7) = 1 q N π if ( N, > , where q N = 28 N for ( N,
7) = 1 , and q N = 7 · (cid:0) (cid:1) N for ( N, > . The in-equality (18) fails when N ≥ . Since the proof for both Ramanujan’s form and Kaplansky’s forms will be the same,for simplicity of notation let L E ( s ) be either L ( E ( − N ) , s ) or L ( E ( − N ) , s ) , and L ( s, χ ) be either L ( s, χ − N ) or L ( s, χ d ) , where d = − N when ( N,
7) = 1 and d = − N when ( N, > . We will get upper bounds for L E (1) and lower boundsfor L (1 , χ ) . Let L n ( s ) = L E ( s + ) . From functional equation of L E ( s ) , we have Λ( s ) = Λ(1 − s ) , where Λ( s ) = q s π − s Γ (cid:18) s + 122 (cid:19) Γ (cid:18) s + 322 (cid:19) L n ( s ) . Since L E ( s ) is L -function of degree 2 and − L ′ E L E ( s ) = ∞ X n =1 λ ( n ) χ ( n ) n s , XPLICIT UPPER BOUNDS FOR L-FUNCTIONS ON THE CRITICAL LINE 11 by Theorem 2.1 choosing λ = 0 . | L E (1) | = log | L n ( ) | (19) ≤ ℜ X n ≤ x λ ( n ) χ ( n ) n . x log n log (cid:0) xn (cid:1) log x + 34 log x log q π + 3 log 24 log x + 32 log x + 8 e − . x / log x . For a lower bound for log | L (1 , χ ) | , we prove the following proposition. Proposition 4.1.
Assume GRH. Let χ be a primitive real character mod q, y ≥ , a ( y ) = y + √ y log y , and b ( y ) = − √ y log y − y . We have log | L (1 , χ ) |≥ X n ≤ y Λ( n ) χ ( n ) n log n log (cid:0) yn (cid:1) log y + 14 log y log q π + log 44 log y − γ y + a ( y ) b ( y ) X n ≤ y Λ( n ) χ ( n ) n log (cid:0) yn (cid:1) log y −
14 log q π − π
24 log y − log 44 + γ . Proof.
Let ρ = + iγ run over the non-trivial zero of L ( s, χ ) . Define F ( s ) = ℜ X ρ s − ρ = X ρ σ − / σ − + ( t − γ ) . From (17) and (18) of Davenport [Da], Chapter 12, if L ( s, χ ) = 0 , (20) − ℜ L ′ L ( s, χ ) = 14 log qπ + 12 ℜ Γ ′ Γ (cid:18) s + 12 (cid:19) − F ( s ) . For s ≥
1, the contribution from the trivial zeros P ∞ k =0 y − k − − s (2 k +1+ s ) ≥
0, and so bythe same arguments as in Lemma (2.4), we have(21) − ℜ L ′ L ( s, χ ) ≥ X n ≤ y Λ( n ) χ ( n ) n s log (cid:0) yn (cid:1) log y + 1log y (cid:18) L ′ L ( s, χ ) (cid:19) ′ + 1log y ℜ X ρ y ρ − s ( ρ − s ) . Integrating (20) as σ varies from 1 to infinity, using (20) and observing that ℜ X ρ Z ∞ y ρ − σ ( ρ − σ ) dσ ≥ − X ρ Z ∞ y / − σ | ρ − | dσ = − X ρ y − / | ρ − | log y = − y − / F (1)log y , we obtain(22)log | L (1 , χ ) | ≥ X n ≤ y Λ( n ) χ ( n ) n log n log (cid:0) yn (cid:1) log y + 14 log y log q π + log 44 log y − γ y − a ( y ) F (1) , where Γ ′ Γ (1) = − γ. To prove the proposition, we need to find the lower bound for − F (1). First notethat taking derivative of (20) at s = 1, we get(23) ℜ (cid:18) L ′ L (1 , χ ) (cid:19) ′ = − (cid:18) Γ ′ Γ (cid:19) ′ (1) − ℜ X ρ − ρ ) ≥ − π − F (1) . From (20),(21) at s = 1 and (23), − F (1) ≥ X n ≤ y Λ( n ) χ ( n ) n log (cid:0) yn (cid:1) log y − π
24 log y − F (1)log y + ℜ y X ρ y ρ − ( ρ − −
14 log qπ + γ ≥ X n ≤ y Λ( n ) χ ( n ) n log (cid:0) yn (cid:1) log y − π
24 log y − F (1)log y − F (1) √ y log y −
14 log q π − log 44 + γ . Therefore, − F (1) ≥ b ( y ) X n ≤ y Λ( n ) χ ( n ) n log (cid:0) yn (cid:1) log y − π
24 log y −
14 log q π − log 44 + γ . Putting the inequality above into (22), we prove the proposition. (cid:3)
To get an upper bound for log | L E (1) || L (1 ,χ ) | , we choose x = 600 in (19) and choose y = 2100 in Proposition 4.1. Thuslog | L E (1) || L (1) | ≤ . . q π + X n = p k p ≤ (cid:0) λ ( n ) n . / log x log n log (cid:0) xn (cid:1) log x − n )log n log (cid:0) yn (cid:1) log y − a ( y ) b ( y )Λ( n ) log (cid:0) yn (cid:1) log y (cid:1) χ ( n ) n + X n = p =601 (cid:12)(cid:12)(cid:12)(cid:12) n + 2 a ( y ) b ( y ) (cid:12)(cid:12)(cid:12)(cid:12) Λ( n ) n log (cid:0) yn (cid:1) log y . To calculate the first sum, we use the exact value of λ ( n ) of L E ( s ) (see [7] section 2for Ramanujan’s form and [8] section 3.2 for Kaplansky’s form). Using computer,for Ramanujan’s form, we get contradiction for (17) when N ≥ · . Similarly,for Kaplansky’s forms, (18) fails when N ≥ · . Appendix: Proof of Lemma 2.3
Proof.
From [1] p.202, we have ℜ Γ ′ Γ ( z ) = log | z | − ℜ z + I ( z ) , where I ( z ) = −ℜ Z ∞ ηη + z · dηe πη − − Z ∞ η ( η + x − y )( η + x − y ) + 4 x y dηe πη − . If y − x ≤ , it is clear that I ( z ) ≤ , and ℜ z ≥
0; hence the lemma is proved.Now we assume y − x > . There are two possiblities for this case.
Case 1: | z | ≥ . To prove the lemma, first we will show that(24) I ( z ) ≤ | z | + 4 e − π | z | / (cid:18) | z | (cid:19) . XPLICIT UPPER BOUNDS FOR L-FUNCTIONS ON THE CRITICAL LINE 13
Integrating I ( z ) by part we obtain −ℜ Z ∞ ηη + z · dηe πη − π ℜ Z ∞ z − η ( η + z ) log(1 − e − πη ) dη ≤ π Z | z | / + Z ∞| z | / | z − η || η + z | log (cid:18) − e − πη (cid:19) dη. In the first integral | z − η | ≤ | z | , | η + z | ≥ | z | , and R ∞| a | log (cid:16) − e − πη (cid:17) dη ≤ π e π | a | . Hence 1 π Z | z | / | z − η || η + z | log (cid:18) − e − πη (cid:19) dη ≤ | z | . In the second integral | z − η || η + z | ≤ | η + z | + | z | | η + z | . Also | η + z | = | z − iη |·| z + iη | ≥ | z | . Therefore1 π Z ∞| z | / | z − η || η + z | log (cid:18) − e − πη (cid:19) dη ≤ e − π | z | / (cid:18) | z | (cid:19) . Combining the inequalities above, we proved (24). Since x ≥ / | z | ≥ , by(24), −ℜ z + I ( z ) ≤ − | z | + 15128 | z | + 4 e − π | z | / (cid:18) | z | (cid:19) ≤ . Case 2: | z | < I ( z ) is greater than or equal tozero when η ≤ p y − x . Hence −ℜ z + I ( z ) ≤ − x x + y ) − Z √ y − x η ( η + x − y )( η + x − y ) + 4 x y dηe πη − . Let f ( x, y ) be the right hand side of the inequality above. Once we show that f ( x, y ) ≤ , the lemma will be proved. Without loss of generality, we can assumethat y ≥ . For any fixed y > x,∂∂x f ( x, y ) = 2 x − y x + y ) + Z √ y − x ηe πη − x ( η + x − y ) + 8 xy ( η − y )(( η + x − y ) + 4 x y ) dη. Since y − x > , the first term is less than 0. Also 2 x ( η + x − y ) +8 xy ( η − y ) < y p y − η > y − η ≥ y − x − η . Therefore f ( x, y ) is decreasing withrespect to x . Because x ≥ /
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