Classification of L-functions of degree 2 and conductor 1
aa r X i v : . [ m a t h . N T ] S e p CLASSIFICATION OF L -FUNCTIONS OF DEGREE AND CONDUCTOR J.KACZOROWSKI and A.PERELLI
Abstract.
We give a full description of the functions F of degree 2 and conductor 1 inthe general framework of the extended Selberg class S ♯ . This is performed by means of a newnumerical invariant χ F , which is easily computed from the data of the functional equation. Weshow that the value of χ F gives a precise description of the nature of F , thus providing a sharpform of the classical converse theorems of Hecke and Maass. In particular, our result confirms,in the special case under consideration, the conjecture that the functions in the Selberg class S are automorphic L -functions. Mathematics Subject Classification (2010):
Keywords: converse theorems; Selberg class; functional equations; modular forms.Contents1. Introduction2. Definitions, notation and basic requisites3. Outline of the proof4. Invariants5. Virtual γ -factors6. Period functions7. Conclusion of the proofReferences 1. Introduction
It is generally expected that the Selberg class S of L -functions coincides with the class ofautomorphic L -functions. This is known for functions of degrees 0 < d <
2, see Conrey-Ghosh [2] and the authors’ papers [9] and [13], but already the important case of degree d = 2appears at present to be out of reach in its full generality. In that case, it is conjecturedthat the primitive functions in S coincide, roughly, with the L -functions of holomorphic andnon-holomorphic eigenforms of any level. Since the Ramanujan conjecture is not known inthe non-holomorphic case, while it is assumed in the definition of S , here we consider theweaker conjecture that the functions of degree 2 in S are contained, roughly, in the family ofthe L -functions of the above eigenforms. We refer to Section 2 for the definitions and notionsintroduced in this section.The latter conjecture is actually known, thanks to the classical converse theorems of Hecke[4], Maass [20], Weil [30], Jacquet-Langlands [5] and the theory of Hecke operators, provided thefunctions F ∈ S satisfy a functional equation with very special Γ-factors and, if the conductor q is greater than 4, certain twists of F by Dirichlet characters also satisfy suitable functionalequations. Concerning general Γ-factors, the only known case of such a conjecture is, as far aswe know, when the conductor q equals 1 and F has a pole at s = 1, in which case it turns outthat necessarily F = ζ ; see [14]. Although the existence of an Euler product expansion for F is not necessary in the converse theorems of Hecke, Maass and Weil, both the Euler productand the pole at s = 1 play a crucial role in the arguments in [14]. In this paper we describe the nature of the functions of degree d = 2 and conductor q = 1in the full generality of the extended Selberg class S ♯ and, as a consequence, we confirm theabove conjecture in the case of conductor 1. Our description of such functions is performed bymeans of a new numerical invariant, denoted χ F and called the eigenweight of F ∈ S ♯ , whichis easy to compute from the data of the functional equation. The name of χ F comes from thefact that it determines the eigenvalue and the weight of the modular form corresponding to F ;see Theorem 1.1. To give a glimpse on our main result, let F ∈ S ♯ with d = 2 and q = 1 benormalized in the sense described below; then, for example, χ F = 0 = ⇒ F ( s ) = ζ ( s ) and χ F = 1212 = ⇒ F ( s ) = L ( s + 112 , ∆) , where L ( s, ∆) is the Ramanujan L -function.In order to state our results we have to introduce a normalization. This is due to the factthat the functional equation in S ♯ reflects F into its conjugate F , while the L -functions ofmodular forms of level 1 are reflected into themselves. We say that F ∈ S ♯ is normalized ifits internal shift θ F vanishes and the first nonvanishing Dirichlet coefficient is 1. Normalizedfunctions have a twofold advantage, making them the right objects for our purpose. Indeed,on one side every F ∈ S ♯ with d = 2 and q = 1 can be normalized by means of a simpleprocedure, see Lemma 4.1, so we may consider only such functions without loosing generality.On the other side, normalized functions have real coefficients, see again Lemma 4.1, and hencethe above mentioned discrepancy disappears.The new invariant χ F is defined as χ F = ξ F + H F (2) + 2 / , (1.1)where ξ F and H F (2) are the H -invariants defined in (2.3)-(2.5) below. Hence χ F can easily becomputed plugging the data of the functional equation of F into the first and second Bernoullipolynomials. For example, if f is a holomorphic cusp form of level 1 and weight k , withfirst nonvanishing Fourier coefficient equal 1, and we denote by L ( s, f ) its L -function, then F ( s ) = L ( s + k − , f ) is a normalized function of S ♯ with d = 2 and q = 1, and χ F = ( k − . Similarly, one easily checks that if u is a Maass form of level 1 and weight 0, with eigenvalue1 / κ and first Fourier-Bessel coefficient equal 1, and L ( s, u ) is its L -function, then F ( s ) = L ( s, u ) is a normalized member of S ♯ with d = 2 and q = 1, and χ F = − κ . Our main result shows, conversely, that the value of χ F is sufficient to detect the nature ofany normalized F ∈ S ♯ of degree 2 and conductor 1. Thus it may be regarded as a sharp formof the Hecke and Maass converse theorems; note that these theorems are used in the very laststep of our proof. Theorem 1.1.
Let F ∈ S ♯ of degree and conductor be normalized. Then χ F ∈ R and(i) if χ F > then there exists a holomorphic cusp form f of level and even integral weight k = 1 + √ χ F such that F ( s ) = L ( s + k − , f ) ;(ii) if χ F = 0 then F ( s ) = ζ ( s ) ;(iii) if χ F < then there exists a Maass form u of level , weight and with eigenvalue / κ = (1 − χ F ) / such that F ( s ) = L ( s, u ) . In case (iii) we can specify the parity ε of u by means of the root number ω F of F , namely ε = 1 − ω F . (1.2)Clearly, if F ∈ S ♯ with d = 2 and q = 1 is not normalized, we may first normalize it and thenuse Theorem 1.1 to detect its nature. Therefore every such F is closely related to one of thethree types of L -functions in Theorem 1.1. Corollary 1.1.
Every F ∈ S ♯ of degree and conductor belongs, modulo normalization,to one of the three families of L -functions in Theorem . . Concerning the functions in S , from the theory of Hecke operators and the linear indepen-dence of Euler products in Kaczorowski-Molteni-Perelli [7],[8] we have the following corollary. Corollary 1.2.
Every F ∈ S of degree and conductor is an automorphic L -function. Remark 1.1.
Since it is well known that exceptional eigenvalues do not exist in level 1, seeSelberg [26], we use this fact in the proof. However, if exceptional eigenvalues would actuallyexist, then our proof would still work, and the resulting theorem would list a second possibilityfor the functions F with χ F <
0, namely F ( s ) = L ( s, u ) for some Maass form u associated withan exceptional eigenvalue. (cid:3) Remark 1.2.
Long ago we conjectured that the functional equation of any F ∈ S ♯ of degree d is completely described by conductor, root number and the H -invariants H F ( n ) with n ≤ d ,see [11, p.103]. This is confirmed by Theorem 1.1 and (1.2) in the present special case where d = 2 and q = 1, since χ F is defined in terms of H F (1) and H F (2); see (1.1) and (2.5). (cid:3) We refer to Section 3 for an outline of the proof, where we describe the ideas leading toTheorem 1.1; here we only add a final remark.
Remark 1.3.
A new and crucial ingredient in the proof of Theorem 1.1 is the fact that,in the present special case, the structural invariants d ℓ ( F ) of F , defined by (2.11) and closelyrelated to the functional equation of F , lie on a certain universal family of algebraic varieties;see Proposition 4.1. We suspect that something similar should hold in general for the functions F ∈ S ♯ , i.e. the d ℓ ( F ) should lie on certain algebraic varieties to a large extent independentof F . In turn, this could explain why L -functions satisfy only functional equations with veryspecial Γ-factors, and in particular could shed some light on the general structure of the Selbergclass. (cid:3) Acknowledgements.
This research was partially supported by the Istituto Nazionale diAlta Matematica, by the MIUR grant PRIN-2015 “Number Theory and Arithmetic Geometry” and by grant 2017/25/B/ST1/00208 “Analytic methods in number theory” from the NationalScience Centre, Poland.2.
Definitions, notation and basic requisites
Throughout the paper we write s = σ + it and f ( s )for f ( s ), f ( s ) ≡ f ( s ) vanishes identically and H denotes the upper half-plane { z = x + iy ∈ C : y > } . The extended Selberg class S ♯ consists of non identically vanishingDirichlet series F ( s ) = ∞ X n =1 a ( n ) n s , absolutely convergent for σ >
1, such that ( s − m F ( s ) is entire of finite order for some integer m ≥
0, and satisfying a functional equation of type F ( s ) γ ( s ) = ωγ (1 − s ) F (1 − s ) , (2.1) where | ω | = 1 and the γ -factor γ ( s ) = Q s r Y j =1 Γ( λ j s + µ j ) (2.2)has Q > r ≥ λ j > ℜ ( µ j ) ≥
0. Note that the conjugate function F has conjugatedcoefficients a ( n ). The Selberg class S is, roughly speaking, the subclass of S ♯ of the functionswith Euler product and satisfying the Ramanujan conjecture a ( n ) ≪ n ε . We refer to Selberg[27], Conrey-Ghosh [2] and to our survey papers [6],[10],[21],[22],[23],[24] for further definitions,examples and the basic theory of the Selberg class.Degree d , conductor q , root number ω F and ξ -invariant ξ F of F ∈ S ♯ are defined by d = d F := 2 r X j =1 λ j , q = q F := (2 π ) d Q r Y j =1 λ λ j j ,ω F = ω r Y j =1 λ − i ℑ ( µ j ) j , ξ F = 2 r X j =1 ( µ j − /
2) := η F + idθ F (2.3)with η F , θ F ∈ R ; θ F is called the internal shift, and the classical L -functions have θ F = 0. Inthis paper we deal only with F ∈ S ♯ of positive degree, hence with r ≥
1. The H -invariants,introduced in [11], are defined for every n ≥ H F ( n ) = 2 r X j =1 B n ( µ j ) λ n − j , (2.4)where B n ( z ) is the n th Bernoulli polynomial. Clearly H F (0) = d, H F (1) = ξ F , H F (2) = 2 r X j =1 µ j − µ j + 1 / λ j . (2.5) Remark 2.1.
Since H -invariants depend only on the data Q, r, λ j , µ j of γ -factors (2.2),we may define such invariants for any γ -factor, without referring to functions F ∈ S ♯ . Moregenerally, the same remark holds for any invariant depending only on the data of γ -factors.Clearly, the invariants of a γ -factor coincide with the corresponding invariants of any F ∈ S ♯ having such a γ -factor. The invariants of γ -factors are usually denoted replacing the suffix F by γ . (cid:3) For α > F ∈ S ♯ is F ( s, α ) = ∞ X n =1 a ( n ) n s e ( − αn /d ) , e ( x ) = e πix , the spectrum of F is defined asSpec( F ) = { α > a ( n α ) = 0 } = n d (cid:0) mq (cid:1) /d : m ∈ N with a ( m ) = 0 o , where n α = qd − d α d and a ( n α ) = 0 if n α N . Moreover, for ℓ = 0 , , . . . we write s ℓ = d + 12 d − ℓd and s ∗ ℓ = s ℓ − iθ F . (2.6) Finally we consider the S -function S F ( s ) := 2 r r Y j =1 sin( π ( λ j s + µ j )) = N X j = − N a j e iπd F ω j s (2.7)with certain N ∈ N , a j ∈ C and − / ω − N < · · · < ω N = 1 /
2, and the h -function h F ( s ) = ω (2 π ) r Q − s r Y j =1 (cid:0) Γ( λ j (1 − s ) + µ j )Γ(1 − λ j s − µ j ) (cid:1) . Here we keep the notation for S F ( s ) used in [18] since we heavily refer to that paper in Section4, but in Section 5 we make a slight change of notation in (2.7), see (5.18). Since the function h F ( s ) depends also on the ω -datum, not present in γ -factors, we define the analog of h F ( s ) for γ -factors as h γ ( s ) = r Y j =1 λ i ℑ ( µ j ) j π ) r Q − s r Y j =1 (cid:0) Γ( λ j (1 − s ) + µ j )Γ(1 − λ j s − µ j ) (cid:1) . (2.8)Instead, the analog S γ ( s ) of S F ( s ) is defined exactly in the same way. Clearly, in view of (2.3),for any γ -factor of F we have that h F ( s ) = ω F h γ ( s ) . (2.9) We recall that every F ∈ S ♯ has polynomial growth on verticalstrips. Moreover, the standard twist F ( s, α ) is entire if α Spec( F ), while for α ∈ Spec( F )it is meromorphic on C with at most simple poles at the points s = s ∗ ℓ , with residue denotedby ρ ℓ ( α ). It is known that ρ ( α ) = 0 for every α ∈ Spec( F ). Further, F ( s, α ) has polynomialgrowth on every vertical strip. We refer to [12] and [15] for these and other results on F ( s, α ).The functions S F ( s ), S γ ( s ), h F ( s ) and h γ ( s ) are invariants, and functional equation (2.1) canbe written in the invariant form F ( s ) = h F ( s ) S F ( s ) F (1 − s ) , (2.10)see Section 1.2 of [18]. Moreover, for large | s | outside an arbitrarily small angular regioncontaining the positive real axis we have the asymptotic expansion h F ( s ) ≈ ω F √ π (cid:18) q /d πd (cid:19) d ( − s ) ∞ X ℓ =0 d ℓ ( F )Γ (cid:0) d ( s ∗ ℓ − s ) (cid:1) , d ( F ) = d idθ F , (2.11)where ≈ means that cutting the sum at ℓ = M one gets a meromorphic remainder which is ≪ than the modulus of the M -th term times 1 / | s | ; see Section 1.3 of [18]. Since (2.11) is provedby means of Stirling’s formula without using the properties of F ∈ S ♯ , see Section 3.2 of [18],a completely analogous expansion holds for h γ ( s ) as well. Clearly, such an expansion holdswithout the factor ω F and with d ℓ ( F ) replaced by d ℓ ( γ ). The coefficients d ℓ ( F ) and d ℓ ( γ ) arecalled the structural invariants and play an important role in the Selberg class theory. When α ∈ Spec( F ), Theorem 3 of [18] shows that the residue ρ ℓ ( α ) of F ( s, α ) at s = s ∗ ℓ is given by ρ ℓ ( α ) = d ℓ ( F ) d ω F √ π e − i π ( ξ F + ds ∗ ℓ ) (cid:18) q /d πd (cid:19) d − ds ∗ ℓ a ( n α ) n − s ∗ ℓ α . If F ∈ S ♯ has d = 2, q = 1 and is normalized, the above quantities simplify as follows: θ F = 0 , Spec( F ) = { √ m : m ∈ N with a ( m ) = 0 } , n α = α / , s ∗ ℓ = s ℓ = 34 − ℓ ,h F ( s ) ≈ ω F √ π (4 π ) s − ∞ X ℓ =0 d ℓ ( F )Γ(3 / − s − ℓ ) , d ( F ) = 1 ,ρ ℓ ( α ) = e iπ/ a ( α / √ α d ℓ ( F )( − πi ) − ℓ α − ℓ for α ∈ Spec( F ) . (2.12)Note that we used Lemma 4.1 in Section 4 to get the above simplified expression of ρ ℓ ( α ). Notealso that Spec( F ) is an infinite set. Indeed, F cannot be a Dirichlet polynomial since, by thefunctional equation, its Lindel¨of function µ F ( σ ) is positive for negative values of σ .3. Outline of the proof
In Section 4 we start the proof with a finer investigation of the properties of the structuralinvariants d ℓ ( F ) with ℓ ≥
0, introduced in [18], in the case of normalized functions F ∈ S ♯ ofdegree 2 and conductor 1. These invariants do not characterize completely the functional equa-tion of F , but contain a good amount of information on it. In particular, the d ℓ ( F ) essentiallydetermine the h -function h F ( s ) appearing in the invariant form (2.10) of the functional equa-tion; see (2.12). The main tool in such investigation is a fully explicit version of a special caseof the transformation formula for nonlinear twists studied in [13],[14],[16],[17]; see Lemma 4.2.From this formula we derive an interesting result in itself, namely: for every integer N ≥ d ℓ ( F ) with ℓ ≤ N lie on the algebraic variety defined by Q N ( X , . . . , X N ) = 0, where Q N are certain universal quadratic forms, i.e. independent of F ; see Proposition 4.1. Since d ( F ) = 1, an inductive argument immediately shows that all d ℓ ( F ) with ℓ ≥ d ( F ) by means of a procedure not depending on F . Then we derive the value of d ( F ) interms of H -invariants, which are easy to compute from the data of the functional equation of F . Actually, recalling (1.1) and (2.5), it turns out that d ( F ) = χ F −
18 ; (3.1)see Lemma 4.3. Moreover, d ( F ) is real-valued in the present case.Next, in Section 5 we introduce the virtual γ -factors γ ( s ) = ( (2 π ) − s Γ( s + µ ) with µ > π − s Γ (cid:0) s + ε + iκ (cid:1) Γ (cid:0) s + ε − iκ (cid:1) with ε ∈ { , } and κ ≥
0, (3.2)respectively of Hecke and Maass type, and consider the analogues h γ ( s ) of the h -function, S γ ( s )of the S -function in (5.18) and d ℓ ( γ ) of the structural invariants, see (5.8). Although not everyvirtual γ -factor corresponds to an existing L -function, the invariants d ℓ ( γ ) satisfy the sameproperties of the d ℓ ( F ); see Lemma 5.2. Moreover, since χ γ := ξ γ + H γ (2) + 23 = ( µ − κ (3.3)with µ and κ as in (3.2), thanks to the analog of (3.1) the set { d ( γ ) : γ virtual γ -factor } coincides with R . Hence to any F we associate a unique virtual γ -factor such that d ( F ) = d ( γ ) . (3.4) Thus d ℓ ( F ) = d ℓ ( γ ) for every ℓ ≥ h F ( s ) = ω F h γ ( s ); see Proposition 5.1. As a consequence, F satisfies the functionalequation γ ( s ) F ( s ) = ω F R ( s ) γ (1 − s ) F (1 − s ) , R ( s ) = S F ( s ) S γ ( s ) , (3.5)where γ is the virtual γ -factor associated with F ; see Corollary 5.1. Now we observe that if R ( s ) = 1, then (3.5) becomes a functional equation of Hecke or Maass type, thus Theorem1.1 follows at once from classical converse theorems, see Lemma 5.1, thanks to (3.1),(3.3) and(3.4).In order to prove that R ( s ) = 1 we first show, under a certain assumption if χ F >
0, that R ( s ) is almost constant, namely as | t | → ∞ R ( s ) = η + O ( e − δ | t | ) with some δ > η = 0; (3.6)see Lemma 5.3. Moreover, if R ( s ) = η then the function S F ( s ) in (5.18), namely S F ( s ) = N X j =0 a j e iπω j s with a j = 0 and ω j strictly increasing , (3.7)has N ≥ a N − e iπω N − s with ω N − >
0; (3.8)see Lemma 5.4.The last step, i.e. proving that actually R ( s ) = η (and hence η = 1), requires deeperarguments and involves a non-standard use of certain period functions , in the sense of Lewis-Zagier [19]. Indeed, given F and its virtual γ -factor, in Section 6 we consider the Fourierseries f ( z ) = ∞ X n =1 a ( n ) n λ e ( nz ) , z ∈ H , where λ = µ or λ = iκ according to (3.2), and the period function ψ ( z ) = f ( z ) − z − λ − f ( − /z ) . Then we proceed to the analysis of the function f ( z ), involving the use of (3.6), of the propertiesof certain Mittag-Leffler functions and of the three-term functional equation (6.22) satisfied by ψ ( z ). This leads to Propositions 6.1 and 6.2, where we show that the assumption in Lemma5.3 holds true and ψ ( z ) = ω F Q η ( z ) + H ( z ) , say, where for some c = c ( λ ) > Q η ( z ) = 12 πi Z ( c ) (cid:0) R ( s − λ ) − η )Γ( s ) γ (1 − s + λ ) γ ( s − λ ) F (1 − s + λ )( − πiz ) − s d s (3.9)and both H ( z ) and Q η ( z ) are holomorphic for | arg( z ) | < π .Finally, in Section 7 we analyse the integral Q η ( z ), plugging into the function R ( s − λ ) in(3.9) the expression (3.7) of S F ( s − λ ) as linear combination of exponentials. It turns out thatthe sum of all resulting terms, but the one coming from (3.8), gives raise to a function H ( z ),say, holomorphic in the sector − π min(1 − ω N − , < arg( z ) < π ;see (7.3)-(7.5). Instead, the term involving (3.8) can be transformed to cf (cid:0) e iπ (1 − ω N − ) z (cid:1) + H ( z ) with a certain function H ( z ) and c = 0, and we obtain that f (cid:0) e iπ (1 − ω N − ) z (cid:1) is holomorphic inthe sector − π min(1 − ω N − , − ω N − + δ, < arg( z ) < πω N − ,δ being as in (3.6); see (7.6)-(7.10). After a change of variable we therefore get that f ( z ) isholomorphic for − δ < arg( z ) < π with some δ >
0. But f ( z ) is 1-periodic, hence it is anentire function; this leads immediately to a contradiction. Thus R ( s ) = η and Theorem 1.1follows. 4. Invariants
We refer to Sections 1 and 2 for definitions and notation used in what follows. We startrecalling the basic properties of normalized functions in S ♯ with degree 2 and conductor 1. Lemma 4.1.
Let F ∈ S ♯ be of degree and conductor . Then F can be normalized. If F is normalized then its Dirichlet coefficients are real, ξ F is an even integer and ω F = − e i π ξ F .Moreover, all its H -invariants are real and its γ -factor satisfies γ ( s ) = γ ( s ) . Proof. If F has a pole at s = 1 then θ F = 0, see Lemma 4.1 of [14], while an entire F canalways be shifted by a purely imaginary quantity to get θ F = 0. Clearly, such a shift doesnot change degree and conductor. Hence, if necessary, every F ∈ S ♯ with d = 2 and q = 1can be normalized by a suitable shift and multiplication by a constant. The second statementfollows from equation (5) and the theorem in [17], observing that ω ∗ F there is denoted here by ω F and H F (1) = ξ F , see (2.5). The assertion about H -invariants follows since the µ -data ofthe conjugate function F are µ j , j = 1 , . . . , r , thus H F ( n ) = H F ( n ) for every n ≥ H F ( n ) ∈ R in our case. This also explains why γ ( s ) = γ ( s ). (cid:3) Since every F ∈ S ♯ with d = 2 and q = 1 can be normalized, from now on we always considernormalized functions, as in Theorem 1.1.A major source of information about invariants comes from the transformation formula fornonlinear twists, which we studied in [13],[14],[16],[17]. Indeed, roughly speaking, the trans-formation formula gives different outputs if the same nonlinear twist of a function F ∈ S ♯ is written in formally different ways; this phenomenon imposes several constraints on the in-variants. Actually, some of the results in Lemma 4.1 were obtained along these lines, and toproceed further we need a closer analysis of the transformation formula. This is done, in aspecial case relevant for our purposes, in the next lemma. For α > F ( s, α ) in the form F ( s ; f ) = ∞ X n =1 a ( n ) n s e ( − f ( n, α )) , f ( n, α ) = n + α √ n. For any integer m ≥ s, α )-variables W m ( s, α ) = ∞ X ν =0 ∞ X µ =0 ∞ X k =3 ν ∞ X ℓ =0 ∞ X h =02 | ( µ + k ) − ν + µ + k +2 ℓ + h = m A ( ν, µ, k, ℓ, h ) × (cid:18) − − s − ℓ µ (cid:19)(cid:18) − s + 2 ν − µ − k − ℓh (cid:19) d ℓ ( F ) α h , (4.1)where d ℓ ( F ) are the structural invariants of F , A ( ν, µ, k, ℓ, h ) = 1 √ πν ! (cid:0) − iπ (cid:1) µ + k (cid:0) − (cid:1) h (4 π ) ν − ℓ a k,ν Γ (cid:0) µ + k + 12 (cid:1) i ν + ℓ (4.2) and the coefficients a k,ν are defined by the expansion (cid:16) ∞ X k =3 (cid:18) / k (cid:19) ξ k (cid:17) ν = ∞ X k =3 ν a k,ν ξ k . (4.3)Note that W m ( s, α ) is a polynomial since k ≥ ν and hence equation − ν + µ + k + 2 ℓ + h = m in (4.1) has only finitely many solutions for every m . Lemma 4.2.
Let F be as in Theorem . and α ∈ Spec( F ) . Then, with the above notation,for every integer M ≥ we have F ( s ; f ) = M X m =0 W m ( s, α ) F (cid:0) s + m , α (cid:1) + H M ( s, α ) , where W ( s, α ) ≡ and the function H M ( s, α ) is holomorphic for σ > − ( M − / . Proof.
We follow closely the arguments in [13] and [16], see also [17], giving details only atthe places which are important for this paper. Moreover, since our goal here is to computeexplicitly the quantities appearing in the main term of the transformation formula, we proceedin an essentially formal way, largely disregarding the problems about convergence, regularityand error terms. Indeed, for these issues a detailed treatment already appears in the abovementioned papers. Therefore, in order to simplify the presentation we denote generically by E any quantity not contributing to our explicit computations, and replace equality by ∼ whenevera finite sum plus error term is replaced by the full expansion. We also correct some minor in-accuracies in our previous computations. Such inaccuracies were unimportant for the results in[13] and [16], while precise computations of the involved quantities are essential here. Moreover,we simply write d ℓ for the structural invariants d ℓ ( F ).(a) Set up.
Let α >
0, not necessarily in Spec( F ). Since in our case f ( n, α ) = n + α √ n = n κ + αn κ , say, as in Section 2 of [16] for X > s ∈ C we write F X ( s ; f ) = ∞ X n =1 a ( n ) n s e ( − f ( n, α )) e − ( n + √ n ) /X . Moreover, as in Section 2.1 of [16], let R ≥ / R N , − R < σ < − R + δ with a small δ > ρ > R + 1) / ∅ 6 = A ⊂ { , } with cardinality |A| , w = w + 12 w , d w = d w d w , z = 1 X + 2 πi, z = 1 X + 2 πiα,G ( w ) = Γ( w )Γ( w ) z − w z − w and let w |A , d w |A and G ( w |A ) be the restriction of w , d w and G ( w ) to the indices appearingin A , respectively. For σ > − R and ℜ ( w ν ) = ρ for ν = 0 , ℜ ( s + w ) >
1, hence byMellin’s transform we get, as in (2.1) of [16], that F X ( s ; f ) = 1(2 πi ) Z ( ρ ) Z ( ρ ) F ( s + w ) G ( w )d w . Next we shift the lines of integration to − η with some 1 / < η < /
4. Writing Z L d w = Z ( − η ) Z ( − η ) d w and analogously for Z L |A d w |A , and I X ( s, A ) = 1(2 πi ) |A| Z L |A F ( s + w |A ) G ( w |A )d w |A , as in Lemma 2.1 of [16] we have that F X ( s ; f ) = X ∅6 = A⊂{ , } I X ( s, A ) + E . (4.4)Now, recalling that the coefficients of F are real by Lemma 4.1, since σ − η < I X ( s, A ) the functional equation of F in the invariant form F ( s + w |A ) = h F ( s + w |A ) S F ( s + w |A ) F (1 − s − w |A ) , (4.5)see (2.10), and expand F (1 − s − w |A ). This way, writing x |A = s + w |A as on p.7656 of [16],we obtain that I X ( s, A ) = ∞ X n =1 a ( n ) n − s πi ) |A| Z L |A h F ( x |A ) S F ( x |A ) G ( w |A ) n w |A d w |A . (4.6)Note that the right hand side of (4.6) is formally slightly different from (2.10) of [16], as here weuse the invariant form (4.5) of the functional equation. Precisely, comparing with the notationin (2.10) of [16] we have S F ( x |A ) = 2 r S ( x |A ) and h F ( x |A ) = ω (2 π ) r Q − x |A e G ( x |A ) . However, this slight difference does not affect the subsequent arguments in [13],[16],[17]. Ac-tually, our present approach simplifies the treatment since we can appeal directly to the as-ymptotic expansion in (2.12) of the function h F ( s ), involving the structural invariants d ℓ . Thepresence of the invariants d ℓ provides also a conceptual advantage, which turns out to be im-portant later on in the paper.(b) Expansions.
From (2.12) of [16] we have that S F ( x |A ) = a − N e − iπ x |A + a N e iπ x |A + E . From the treatment on pages 7657–7659 of [16] of the J X -integrals, stemming from the term a N e πi x |A , we have that in the end their contribution is absorbed by the function H M ( s, α ) inthe statement of the lemma; see also Lemma 2.2 of [16]. Thus the term a N e πi x |A does notcontribute to the main terms, hence we rewrite the above expression for S F ( x |A ) as S F ( x |A ) = a − N e − iπ x |A + E . Next we recall that a − N = e − i π ξ F , see Section 3.7 of [18], hence ω F a − N = − h F ( s ) in (2.12) we have that equation (2.13) of [16] isnow replaced by h F ( x |A ) S F ( x |A ) ∼ − (4 π ) x |A π ) / e − iπ x |A ∞ X ℓ =0 d ℓ Γ(3 / − x |A − ℓ ) + E . (4.7)Thus, by (4.6) and (4.7), equation (2.15) of [16] finally becomes I X ( s, A ) ∼ − (4 π ) s π ) / ∞ X ℓ =0 d ℓ ∞ X n =1 a ( n ) n − s I X ( s, A , n, ℓ ) + E (4.8)with I X ( s, A , n, ℓ ) = 1(2 πi ) |A| Z L |A Γ(3 / − x |A − ℓ (cid:1) e − iπ x |A G ( w |A ) (cid:0) (4 π ) n (cid:1) w |A d w |A . Note that I X ( s, A , n, ℓ ) coincides with the definition of I X ( s, A , n, ℓ ) in (2.16) of [16] with d = 2, q = 1 / (4 π ) and θ F = 0, but not with the definition of I X ( s, A , n, ℓ ) on p.1409 of [13]. Indeed,recalling the data of the functional equation of F in Section 2, the latter coincides with (cid:0) β (cid:1) s I X ( s, A , n, ℓ ) , where β = r Y j =1 λ λ j j , (4.9)since Q β/d d = (4 π ) − in the present case, where degree is 2 and conductor is 1.Now we use the Mellin transform argument leading to equation (2.18) of [16], but in orderto compute the coefficients f ℓ in such equation we have to follow the arguments on p.1410 of[13]. Recalling the difference in notation pointed out in (4.9), by (2.13) of [13] we have thatthe inverse Mellin transform e I X ( y ) of (cid:0) β (cid:1) s I X ( s, A , n, ℓ ) is in our case e I X ( y ) = 12 e i/ √ yβ (cid:16) e − iπ/ √ yβ (cid:17) − ℓ Y ν ∈A (cid:16) e − zνQ κνnκν yκν − (cid:17) . Hence, after a computation needed since the constants c ℓ in (2.13) and (2.14) of [13] are notthe same, equation (2.14) of [13] in the present case becomes (cid:0) β (cid:1) s I X ( s, A , n, ℓ ) = (cid:0) β (cid:1) s c ℓ Z ∞ e i √ x Y ν ∈A (cid:16) e − z ν ( x (4 π )2 n ) κν − (cid:17) x − s − ℓ − d x, where c ℓ = 12 e − πi i ℓ . (4.10)As a consequence, using identity (2.16) of [13], equation (2.19) of [16] now takes the form I X ( s, n, ℓ ) := X ∅6 = A⊂{ , } I X ( s, A , n, ℓ )= c ℓ Z ∞ e i √ x (cid:0) e − Ψ X ( x,n ) e (cid:0) − f (cid:0) x (4 π ) n , α (cid:1)(cid:1) − (cid:17) x − s − ℓ − d x (4.11)with Ψ X ( x, n ) = 1 X (cid:16) x (4 π ) n + r x (4 π ) n (cid:17) . (c) Saddle point and limit as X → ∞ . The next step is to apply the saddle point techniquedeveloped in Section 2.3 of [13] and [16] to the right hand side of (4.11). Recalling that θ F = 0and that the integral on the right hand side of (4.11) is multiplied by c ℓ , from Lemma 2.3 of[16] we deduce that for n ≥ n and n sufficiently large I X ( s, n, ℓ ) = c ℓ K X (cid:0) s + ℓ , n (cid:1) + E , (4.12)where K X ( s, ξ ) is as in (2.21) of [16]. Hence from (4.4),(4.8),(4.10) and (4.12) we have that(2.22) of [16] becomes F X ( s ; f ) ∼ e iπ/ π ) / (4 π ) s ∞ X ℓ =0 i ℓ d ℓ X n ≥ n a ( n ) n − s K X ( s + ℓ , n ) + E Then we pass to the limit as X → ∞ as in Section 2.4 of [16], thus getting that F ( s ; f ) ∼ e iπ/ π ) / (4 π ) s ∞ X ℓ =0 i ℓ d ℓ X n ≥ n a ( n ) n − s K ( s + ℓ , n ) + E , (4.13) where K ( s, n ) = γx − s Z r − r e i Φ( z,n ) (1 + γλ ) − s − d λ and x = x ( n ) is the critical point of Φ( z, n ), see Lemma 2.3 of [13],Φ( z, n ) = z − πf (cid:0) z (4 π ) n , α (cid:1) , z = x (1 + γλ ) ,γ = 1 − i, r = log n √ R , R = x Φ ′′ ( x , n ) . We finally note that the critical point x is real andlog n √ n ≪ r ≪ log n √ n , (4.14)see Lemma 2.3 and (2.30) of [13], respectively. Note also that, with respect to [13] and [16], weslightly simplified the notation of the various terms in K ( s, ξ ).(d) Computation of K ( s, n ) . Now we proceed to the novel part of the proof of the lemma,namely a detailed computation of the function K ( s, n ). From now on we assume that α ∈ Spec( F ), thus α / ∈ N . Clearly, the critical point x , solution of the equation ∂∂z Φ( z, n ) = 0,in our case satisfies √ x = 4 πn ∆ n with ∆ n = 1 − α √ n , (4.15)hence Φ( x , n ) = 2 π (cid:0) n − α √ n + α (cid:1) and e i Φ( x ,n ) = e ( − α √ n ) . Therefore K ( s, n ) = γx − s e i Φ( x ,n ) Z r − r e i (Φ( z,n ) − Φ( x ,n )) (1 + γλ ) − s − d λ = γe ( − α √ n )(4 πn ∆ n ) − s I ( s, n ) , (4.16)where I ( s, n ) = Z r − r e i (Φ( z,n ) − Φ( x ,n )) (1 + γλ ) − s − d λ. But Φ( z, n ) − Φ( x , n ) = √ z ∆ n − z πn − √ x ∆ n + x πn = 4 πn ∆ n (cid:0) (1 + γλ ) / − − γλ (cid:1) , hence writing φ ( x ) = (1 + x ) / − − x + x = ∞ X k =3 (cid:18) / k (cid:19) x k (4.17)and recalling that γ = − i we have that I ( s, n ) = Z r − r e πin ∆ n ( φ ( γλ )+ iλ ) (1 + γλ ) − s − d λ = Z r − r e − πn ∆ n λ e (cid:0) n ∆ n φ ( γλ ) (cid:1) (1 + γλ ) − s − d λ. (4.18) Since r = o (1) as n → ∞ by (4.14), for n ≥ n we expand the complex exponential andthe function (1 + γλ ) − s − in the above integral, and then we replace φ ( γλ ) by its power series.Hence we have e (cid:0) n ∆ n φ ( γλ ) (cid:1) (1 + γλ ) − s − = ∞ X ν =0 (cid:0) πin ∆ n φ ( γλ ) (cid:1) ν ν ! ∞ X µ =0 (cid:18) − s − µ (cid:19) ( γλ ) µ and by (4.17) and (4.3) we write φ ( γλ ) ν = ∞ X k =3 ν a k,ν ( γλ ) k with a , = 1 . Thus, thanks to the symmetry of the integral, (4.18) becomes I ( s, n ) = 2 ∞ X ν =0 ∞ X µ =0 ∞ X k =3 ν | ( µ + k ) (4 πin ∆ n ) ν ν ! (cid:18) − s − µ (cid:19) a k,ν γ µ + k Z r e − πn ∆ n λ λ µ + k d λ. (4.19)But, first by the change of variable √ n ∆ n λ = u , then completing the integral to (0 , ∞ ) andfinally by the change of variable πu = ξ , in view of (4.14) we have Z r e − πn ∆ n λ λ µ + k d λ = 1( √ n ∆ n ) µ + k +1 Z ∞ e − πu u µ + k d u + E = 12( √ πn ∆ n ) µ + k +1 Z ∞ e − ξ ξ µ + k − d ξ + E = Γ (cid:0) µ + k +12 (cid:1) √ πn ∆ n ) µ + k +1 + E . Hence (4.19) becomes I ( s, n ) = ∞ X ν =0 ∞ X µ =0 ∞ X k =3 ν | ( µ + k ) (4 πi ) ν ν ! a k,ν γ µ + k Γ (cid:0) µ + k +12 (cid:1) π µ + k +12 (cid:18) − s − µ (cid:19) ∆ ν − µ − k − n n − ν − µ − k − + E , (4.20)and from (4.16) and (4.20) we obtain that K ( s, n ) = γe ( − α √ n )(4 π ) − s ( n ∆ n ) − s × ∞ X ν =0 ∞ X µ =0 ∞ X k =3 ν | ( µ + k ) (4 πi ) ν ν ! a k,ν γ µ + k Γ (cid:0) µ + k +12 (cid:1) π µ + k +12 (cid:18) − s − µ (cid:19) ∆ ν − µ − k − n n − ν − µ − k − + E = ∞ X ν =0 ∞ X µ =0 ∞ X k =3 ν | ( µ + k ) b ν,µ,k ( s ) ∆ ν − µ − k + − sn n s − − ν − µ − k e ( − α √ n ) + E , (4.21)where b ν,µ,k ( s ) = (4 π ) − s (4 πi ) ν ν ! a k,ν γ µ + k +1 Γ (cid:0) µ + k +12 (cid:1) π µ + k +12 (cid:18) − s − µ (cid:19) . Note that b ν,µ,k ( s ) is independent of α , n and the data of the functional equation of F .Recalling the definition of ∆ n in (4.15) we have the following expansion∆ ν − µ − k + − sn = ∞ X h =0 (cid:18) ν − µ − k + − sh (cid:19) ( − h α h h n h/ , hence from (4.13) and (4.21) (with s replaced by s + ℓ/
2) we have that F ( s ; f ) ∼ ∞ X ν =0 ∞ X µ =0 ∞ X k =3 ν ∞ X ℓ =0 ∞ X h =02 | ( µ + k ) c ν,µ,k,ℓ,h ( s ) d ℓ α h X n ≥ n a ( n ) n s + ℓ − ν − µ − k − h e ( − α √ n ) + E (4.22)with c ν,µ,k,ℓ,h ( s ) = e iπ/ π ) / (4 π ) s i ℓ (cid:18) ν − µ − k + − s − ℓh (cid:19) ( − h h b ν,µ,k (cid:0) s + ℓ (cid:1) . (4.23)Note that, again, the coefficients c ν,µ,k,ℓ,h ( s ) are independent of α , n and the data of the func-tional equation of F . Hence, recalling that γ = 1 − i and rearranging terms, a comparison of(4.22) and (4.23) with (4.1) and (4.2) shows that (4.22) can be rewritten as F ( s ; f ) ∼ ∞ X m =0 W m ( s, α ) X n ≥ n a ( n ) n s + m e ( − α √ n ) + E , where W m ( s, α ) is defined by (4.1). Thus, adding and subtracting the terms with 1 ≤ n < n ,recalling the definition of the standard twist of F we finally get F ( s ; f ) ∼ ∞ X m =0 W m ( s, α ) F (cid:0) s + m , α (cid:1) + E . (4.24)Moreover, from (4.1) we have that W ( s, α ) = A (0 , , , ,
0) = Γ(1 / √ π = 1 . (4.25)Finally, as in the general case treated in [13] and [16], the explicit meaning of (4.24) is thatfor every M ≥ K = K ( M ) → ∞ as M → ∞ such that F ( s ; f ) = M X m =0 W m ( s, α ) F (cid:0) s + m , α (cid:1) + H M ( s, α ) (4.26)holds with a function H M ( s, α ) holomorphic for σ > − K . Given M , if K ( M ) < ( M − / M ∗ > M such that K ( M ∗ ) ≥ ( M − /
2, thus getting that (4.26) holds with M ∗ inplace of M and − ( M − / − K . But for M < m ≤ M ∗ and σ > − ( M − / ℜ ( s + m/ > (1 − M ) / M + 1) / W m ( s, α ) F ( s + m , α ) is holomorphic for σ > (1 − M ) / M < m ≤ M ∗ to the right hand side. (cid:3) The previous lemma is the first step in the proof of the main result of this section.
Proposition 4.1.
For every N ≥ there exists a quadratic form Q N ( X , . . . , X N ) = X ℓ,h ≥ ℓ + h ≤ N α ℓ,h X ℓ X h with α ℓ,h ∈ R and α ,N + a N, = 1 , such that for every F as in Theorem . we have Q N (cid:0) d ( F ) , . . . , d N ( F ) (cid:1) = 0 . Before normalization of coefficients, such a quadratic form is given by (4.34) and (4.29) below. Proof.
Again we write d ℓ for d ℓ ( F ). Thanks to the periodicity of the complex exponentialwe have that the nonlinear twist F ( s ; f ) defined before Lemma 4.2 coincides with the standardtwist F ( s, α ), hence Lemma 4.2 implies that for α ∈ Spec( F ) and M ≥ M X m =1 W m ( s, α ) F (cid:0) s + m , α (cid:1) is holomorphic for σ > (1 − M ) / s = s M = 3 / − M/ M X m =1 W m ( s M , α ) ρ M − m ( α ) = 0 . Hence, thanks to the expression of ρ ℓ ( α ) in (2.12) and recalling that a ( α / = 0, we finallyobtain that for every α ∈ Spec( F ) and M ≥ M X m =1 W m ( s M , α )( − πi ) m d M − m α m = 0 . (4.27)We set k = 3 ν + b with b ≥ ℓ + h is0 ≤ ℓ + h ≤ m − ( ν + µ + b ) ≤ m. Then, rearranging terms, we rewrite W m ( s, α ) as W m ( s, α ) = ∞ X ν =0 ∞ X µ =0 ∞ X b =0 ∞ X ℓ =0 ∞ X h =02 | ( ν + µ + b ) ν + µ + b +2 ℓ + h = m A ( ν, µ, ν + b, ℓ, h ) × (cid:18) − − s − ℓ µ (cid:19)(cid:18) − s − ν − µ − b − ℓh (cid:19) d ℓ ( F ) α h = ∞ X ℓ =0 ∞ X h =02 ℓ + h ≤ m B m ( s, ℓ, h ) d ℓ ( F ) α h , (4.28)where B m ( s, ℓ, h ) = ∞ X ν =0 ∞ X µ =0 ∞ X b =02 | ( ν + µ + b ) ν + µ + b = m − (2 ℓ + h ) A ( ν, µ, ν + b, ℓ, h ) × (cid:18) − − s − ℓ µ (cid:19)(cid:18) − s − ν − µ − b − ℓh (cid:19) . (4.29)Thus from (4.27) and (4.28) we have, still for every α ∈ Spec( F ) and M ≥
1, that M X m =1 ( − πi ) m ∞ X ℓ =0 ∞ X h =02 ℓ + h ≤ m B m ( s M , ℓ, h ) d ℓ d M − m α m + h = 0 . (4.30) Next we set p = m + h , hence (4.30) becomes0 = M X m =1 ( − πi ) m ∞ X ℓ =0 ∞ X p = m ℓ + h ≤ m B m ( s M , ℓ, p − m ) d ℓ d M − m α p = X ≤ m ≤ M ( − πi ) m X m ≤ p ≤ m X ≤ ℓ ≤ m − p/ B m ( s M , ℓ, p − m ) d ℓ d M − m α p = X ≤ p ≤ M (cid:16) X p/ ≤ m ≤ min( p,M ) . X ≤ ℓ ≤ m − p/ ( − πi ) m B m ( s M , ℓ, p − m ) d ℓ d M − m (cid:17) α p (4.31)for every α ∈ Spec( F ) and M ≥ F ) is an infinite set, see at the end of Section 2, thus (4.31) impliesthat for every M ≥ ≤ p ≤ M X p/ ≤ m ≤ min( p,M ) . X ≤ ℓ ≤ m − p/ ( − πi ) m B m ( s M , ℓ, p − m ) d ℓ d M − m = 0 . (4.32)Setting further h = M − m , and hence m = M − h , we see that condition p/ ≤ m ≤ min( p, M )in (4.32) is equivalent to M − min( p, M ) ≤ h ≤ M − p/ M − p, ≤ h ≤ M − p/
2. Therefore we rewrite (4.32) as X max( M − p, ≤ h ≤ M − p/ X ≤ ℓ ≤ M − h − p/ ( − πi ) − h B M − h ( s M , ℓ, p + h − M ) d ℓ d h = 0 (4.33)for every M ≥ ≤ p ≤ M . Finally, we choose p = M and M = 2 N in (4.33), thusgetting (after reversing summation) that for N ≥ e Q N ( d , . . . , d N ) = X ≤ ℓ ≤ N X ≤ h ≤ N − ℓ ( − πi ) − h B N − h ( s N , ℓ, h ) d ℓ d h = 0 . (4.34)This is, before normalization, the quadratic form announced in the proposition.Hence it remains to check our assertions about the quadratic form e Q N ( X , . . . , X N ), namelythat its coefficients are independent of F and real after normalization, and for N ≥ − πi ) − N B N ( s N , , N ) + B N ( s N , N, = 0 . The first assertion is immediate since the coefficients A ( ν, µ, k, ℓ, h ) in (4.2), and hence thepolynomials B m ( s, ℓ, h ) in (4.29), are independent of F , and so is also s N by (2.6). Moreover,in view of (4.2) and (4.29), the coefficients ( − πi ) − h B N − h ( s N , ℓ, h ) are sums of type ∞ X ν =0 ∞ X µ =0 ∞ X b =0 ν + µ + b =2( N − ℓ − h ) i h + ν + µ + b + ν + ℓ c N ( ν, µ, b, ℓ, h ) = i N ∞ X ν =0 ∞ X µ =0 ∞ X b =0 ν + µ + b =2( N − ℓ − h ) ( − ν c N ( ν, µ, b, ℓ, h )with c N ( ν, µ, b, ℓ, h ) ∈ R , so the second assertion follows. Finally, again from (4.2) and (4.29)we have that B N ( s N , , N ) = A (0 , , , , N ) (cid:18) − s N N (cid:19) = ( − − N (cid:18) N − N (cid:19) and B N ( s N , N,
0) = A (0 , , , N,
0) = 1 √ π (4 π ) − N Γ(1 / i N = ( − πi ) − N , thus ( − πi ) − N B N ( s N , , N ) + B N ( s N , N,
0) = ( − πi ) − N (cid:16) − N (cid:18) N − N (cid:19)(cid:17) = 0 for N ≥
2, as required. The proposition now follows dividing e Q N by the right hand side of thelast equation. (cid:3) As an immediate consequence of Proposition 4.1 we explicitly record the following corollary,obtained by a trivial induction since d ( F ) = 1, see (2.12). Corollary 4.1.
Let F be as in Theorem . . Then the value of any d ℓ ( F ) with ℓ ≥ isdetermined by the value of d ( F ) by a recursive algorithm independent of F . Finally we express the structural invariant d ( F ) in terms of the H -invariants, which are easyto compute by means of definition (2.4). Lemma 4.3.
Let F be as in Theorem . . Then, recalling (1.1) , we have that d ( F ) = χ F − . Proof.
We briefly sketch the formal computations leading to the result; actually, thesecomputations lead to an expression of any d ℓ ( F ) in terms of H -invariants, see (4.39), which canbe used to calculate explicitly any given d ℓ ( F ).Using (2.3) and Stirling’s formula, see Corollary 6.2 of [29], adopting the notation ∼ in theproof of Lemma 4.2 we getlog h F ( s ) − log Γ(3 / − s ) ∼ log ω F − log √ π + (2 s −
1) log(4 π )+ ∞ X ν =1 ( − ν +1 ν ( ν + 1) n r X j =1 (cid:16) B ν +1 ( λ j + µ j )( − λ j s ) ν + B ν +1 (1 − µ j )( − λ j s ) ν (cid:17) − B ν +1 (3 / − s ) ν o ∼ log (cid:16) ω F √ π (cid:0) π ) − s (cid:17) + ∞ X ν =1 r ν ( F ) ν ( ν + 1) 1 s ν , say, since d = 2, q = 1 and θ F = 0. Moreover, thanks to (2.4) and to the formulae B ν +1 ( λ j + µ j ) = ν +1 X k =0 (cid:18) ν + 1 k (cid:19) B k ( µ j ) λ ν +1 − kj and B ν +1 (1 − µ j ) = ( − ν +1 B ν +1 ( µ j ) , see (4.23) and (4.27) of [29], we have that r ν ( F ) = B ν +1 (3 / ν − n ν +1 X k =0 (cid:18) ν + 1 k (cid:19) H F ( k ) + ( − ν +1 H F ( ν + 1) o . (4.35)Thus h F ( s ) ∼ ω F √ π (cid:0) π ) − s Γ(3 / − s ) exp (cid:16) ∞ X ν =1 r ν ( F ) ν ( ν + 1) 1 s ν (cid:17) . (4.36)Next, expanding the exponential we obtainexp (cid:16) ∞ X ν =1 r ν ( F ) ν ( ν + 1) 1 s ν (cid:17) ∼ ∞ X m =1 m ! (cid:16) ∞ X ν =1 r ν ( F ) ν ( ν + 1) 1 s ν (cid:17) m ∼ ∞ X h =1 V h ( F ) s h with V h ( F ) = h X m =1 m ! X ν ≥ · · · X ν m ≥ ν + ··· + ν m = h m Y j =1 r ν j ( F ) ν j ( ν j + 1) for h ≥ . (4.37) Moreover, letting z = 1 / − s , for h ≥ s h = X ℓ ≥ h A h,ℓ z ( z − · · · ( z − ℓ + 1) (4.38)with certain coefficients A h,ℓ ∈ R . But, by the factorial formula for the Γ function, for ℓ ≥ / − s ) z ( z − · · · ( z − ℓ + 1) = Γ( z − ℓ + 1) = Γ(3 / − s − ℓ ) , hence substituting into (4.36) and comparing with (2.12) we finally obtain that for ℓ ≥ d ℓ ( F ) = ℓ X h =1 A h,ℓ V h ( F ) . (4.39)In particular, (4.37) and (4.38) imply that for ℓ = 1 we have A , = − V ( F ) = r ( F ) / d ( F ) = − r ( F ) = − B (3 /
2) + 12 (cid:16) H F (0) + 2 H F (1) + 2 H F (2) (cid:17) = 1324 + ξ F + H F (2) , and the result follows in view of (1.1). (cid:3) Virtual γ -factors With the cases of holomorphic and non-holomorphic modular forms of level 1 in mind, wedefine the virtual γ -factors as γ ( s ) = ( (2 π ) − s Γ( s + µ ) with µ > π − s Γ (cid:0) s + ε + iκ (cid:1) Γ (cid:0) s + ε − iκ (cid:1) with ε ∈ { , } and κ ≥
0. (5.1)Obviously, we say that the first γ -factor in (5.1) is of Hecke type and the second is of
Maasstype . In view of Remark 2.1, the virtual γ -factors have degree 2 and conductor 1. Moreover,by (2.5) we also have ξ γ = ( µ − ε −
1) and χ γ := ξ γ + H γ (2) + 23 = ( µ − κ , (5.2)and clearly, with obvious notation, r Y j =1 λ i ℑ µ j j = 1 (5.3)for all virtual γ -factors.As suggested by the name, not every virtual γ -factor corresponds to an existing L -function.The list of all normalized F ∈ S ♯ whose γ -factor is a virtual γ -factor is provided by theclassical converse theorems of Hecke [4] and Maass [20]. In the next lemma, “eigenvalue of theLaplacian” means eigenvalue of the hyperbolic Laplacian on the real analytic, L and Γ (1)-invariant functions on the upper half-plane H . Lemma 5.1.
If the γ -factor of F ∈ S ♯ is a virtual γ -factor and F is normalized, then one ofthe following cases holds true:(i) µ = k − with an even integer k ≥ , in which case ω F = ( − k/ and F ( s ) = L ( s + µ, f ) with some holomorphic cusp form f of level and weight k ;(ii) ρ = 1 / κ is an eigenvalue of the Laplacian, in which case ω F = ( − ε and F ( s ) = L ( s, u ) with some Maass form u of level , weight , parity ε and with eigenvalue ρ . Proof.
We deal first with virtual γ -factors of Hecke type. Suppose that a normalized F ∈ S ♯ satisfies γ ( s ) F ( s ) = ωγ (1 − s ) F (1 − s ) . (5.4)Then a ( n ) ≪ n c for some c > s s − µ , (5.4) becomes(2 π ) − s Γ( s ) G ( s ) = ω (2 π ) − ( k − s ) Γ( k − s ) G ( k − s ) , where k = 2 µ + 1 > G ( s ) = F ( s − µ ) . This is Hecke’s functional equation with signature (1 , k, ω ), see Definition 2.1 of Berndt-Knopp[1]. Moreover, by (5.3) we have ω F = ω. (5.5)But, by Lemma 4.1 and (5.2), ξ F = 2 µ − k is also an even integer and ω = ( − k/ by (5.5). Hence G ( s ) has signature(1 , k, ( − k/ ) with an even integer k ≥ , therefore by Hecke’s converse theorem, see Theorem 2.1 of [1], G ( s ) = L ( s, f ), the L -functionassociated with a modular form f of level 1 and weight k . According to Theorem 4 of ChapterVII of Serre [28] (where the weight of f is denoted by 2 k ) the space of modular forms of weight k ∈ Z is trivial for k < k = 2, while it has dimension 1 for k = 0, 4, 6, 8, 10 and isgenerated, respectively, by 1 and the Eisenstein series G , G , G , G . Moreover, for k ≥ ≥
1. But L ( s, G k ) has a pole at s = k , thus F should have a pole at s = ( k + 1) / = 1, impossible for k ≥
4. On the other hand, if f is a Hecke eigenform of weight k ≥
12 then F ( s ) = L ( s + k − , f ) is normalized, and our firstassertion follows.Next we deal with virtual γ -factors of Maass type. The argument is similar, this time basedon the version allowing poles of Maass’ converse theorem given by Theorem 2.1 of Raghunathan[25], with ν = 1 / iκ . Suppose that a normalized F satisfies (5.4) with such a γ -factor. Thenby (5.5), Lemma 4.1 and (5.2) we have ω = ( − ε . Hence our assertion follows from the abovequoted version of Maass’ converse theorem, since ν (1 − ν ) = 1 / κ = ρ must be an eigenvalueof the Laplacian, thus its eigenspace is nontrivial and therefore we may choose a Maass form u associated with ρ and with parity ε such that F ( s ) = L ( s, u ) is normalized. (cid:3) Remark 5.1.
The limitations µ > κ ≥ γ -factors in (5.1) areactually natural. Indeed, we may restrict to κ ≥ (1), see Selberg [26]. Moreover, if µ ≤ k inthe above proof is an even integer ≤
0, thus by the above quoted Theorem 4 of [28] the spaceof modular forms is either trivial (if k <
0) or generated by 1 (if k = 0), in which case L ( s, F ( s ) ≡
0, impossible. (cid:3)
By (2.8), given any virtual γ -factor in (5.1) we define the virtual h -function as h γ ( s ) = ( (2 π ) s − Γ(1 − s + µ )Γ(1 − s − µ ) π s − Γ (cid:16) − s + ε − iκ (cid:17) Γ (cid:16) − s − ε + iκ (cid:17) Γ (cid:16) − s + ε + iκ (cid:17) Γ (cid:16) − s − ε − iκ (cid:17) . (5.6)Hence by the reflection formula for the Γ function we have that h γ ( s ) S γ ( s ) = γ (1 − s ) γ ( s ) , (5.7)and by (2.12) and (2.9) h γ ( s ) ≈ (4 π ) s − √ π ∞ X ℓ =0 d ℓ ( γ )Γ(3 / − s − ℓ ) , d ( γ ) = 1 . (5.8) Remark 5.2.
Since ε ∈ { , } , the virtual h -functions do not depend on ε thanks tothe symmetry of the Γ-factors in (5.6). In particular, this shows that the h -functions do notdetermine uniquely the γ -factors. (cid:3) Although virtual γ -factors do not correspond, in general, to some F ∈ S ♯ , their structuralinvariants d ℓ ( γ ) satisfy the same properties of the invariants d ℓ ( F ) of the normalized functions F ∈ S ♯ of degree 2 and conductor 1. Lemma 5.2.
The structural invariants d ℓ ( γ ) of virtual γ -factors satisfy the same propertiesof the invariants d ℓ ( F ) in Proposition . , Corollary . and Lemma . . Proof.
We first show that there exist polynomials P ℓ , Q ℓ ∈ R [ x ] such that d ℓ ( γ ) = ( P ℓ ( µ ) Q ℓ ( κ ) (5.9)for every virtual γ -factor in (5.1). Note, in view of Remark 5.2, that the polynomials Q ℓ , if theyexist, do not depend on ε . Now we observe that Lemma 4.3 is proved by Stirling’s formula,without using the properties of F ∈ S ♯ . Hence in the present case (4.39) becomes d ℓ ( γ ) = ℓ X h =1 A h,ℓ V h ( γ ) , (5.10)where A h,ℓ is defined by (4.38) and V h ( γ ) by (4.37) and (4.35), clearly with γ in place of F .But the H -invariants in (4.35) are defined in terms of Bernoulli polynomials involving µ and κ , hence the dependence of d ℓ ( γ ) on µ and κ is polynomial as well. Moreover, A h,ℓ ∈ R so P ℓ ∈ R [ x ], and Q ℓ ( κ ) involves Bernoulli polynomials at iκ and − iκ , thus its coefficients are alsoreal. Hence (5.9) is proved.Next we observe that if a virtual γ -factor is actually a γ -factor of a function F ∈ S ♯ as inTheorem 1.1, then by Proposition 4.1 we have Q N ( d ( γ ) , . . . , d N ( γ )) = 0 . (5.11)But Q N ( d ( γ ) , . . . , d N ( γ )) is a polynomial in µ or κ by (5.9), and by Lemma 5.1 there existinfinitely many µ and κ for which (5.11) holds (remember that the spectrum of the Laplacianis infinite). Thus (5.11) holds identically in µ and κ , i.e. Proposition 4.1 holds for all virtual γ -factors. Hence the analog of Corollary 4.1 holds as well, and the analog of Lemma 4.3 followsfrom (5.10). (cid:3) Now we are ready to prove the main result of this section.
Proposition 5.1.
Let F ∈ S ♯ be as in Theorem . . Then there exists a virtual γ -factorsuch that χ F = χ γ and h F ( s ) = ω F h γ ( s ) . Moreover, such a γ -factor is uniquely determined if we choose ε in (5.1) satisfying ω F = ( − ε ,and it is of Hecke or Maass type depending on χ F > or χ F ≤ , respectively. Proof.
Thanks to Lemmas 4.3 and 5.2 we have d ( F ) = d ( γ ) ⇐⇒ χ F = χ γ . (5.12)Moreover, by Lemmas 4.1 and 4.3 we have d ( F ) ∈ R (recall that ξ F = H F (1)), and by(5.2) the set of values of χ γ coincides with R . Thus there exists a virtual γ -factor such that d ( F ) = d ( γ ). But then, thanks to Corollary 4.1 and Lemma 5.2, we have that d ℓ ( F ) = d ℓ ( γ ) for every ℓ ≥ In view of (5.2) and the equality in the right hand side of (5.12), if χ F > γ -factor of Hecke type. Instead, if χ F ≤ γ -factors of Maass type, depending on the value of ε . Thus, since ω F = ± ε ∈ { , } in (5.1) in such a way that ω F = ( − ε , and the secondassertion follows as well. (cid:3) Recalling the definition of the functions S F ( s ) and S γ ( s ) in Section 2.1 and writing R ( s ) = R F,γ ( s ) := S F ( s ) S γ ( s ) (5.13)we have the following important consequence of Proposition 5.1. Corollary 5.1.
Let F ∈ S ♯ be as in Theorem . and R ( s ) be as in (5.13) . Then there existsa unique virtual γ -factor, in the sense of Proposition . , such that F satisfies the functionalequation γ ( s ) F ( s ) = ω F R ( s ) γ (1 − s ) F (1 − s ) . Proof.
We start with the functional equation of F written as in (2.10) and then use Lemma4.1, Proposition 5.1 and (5.7) to get F ( s ) = ω F h γ ( s ) S F ( s ) F (1 − s ) = ω F γ (1 − s ) γ ( s ) S F ( s ) S γ ( s ) F (1 − s );thus the result follows. (cid:3) We conclude the section with a study of the function R ( s ) in (5.13). We shall always assumethat γ in (5.13) is the virtual γ -factor associated with F in the sense of Proposition 5.1. Lemma 5.3.
Let F be as in Theorem . , R ( s ) be as in (5.13) and assume that µ + 1 / isa positive integer, µ being as in (5.1) . Then as | t | → ∞ R ( s ) = η + O (cid:0) e − δ | t | (cid:1) with some δ > , where η = ( ω F e − i π (2 µ +1) ∈ {− , } if γ is of Hecke type if γ is of Maass type . Proof.
A computation based on (2.7) shows that as | t | → ∞ S F ( s ) = e − sgn( t ) πi ( s + ξ F / (cid:0) O ( e − c | t | ) (cid:1) with some c >
0. Hence by Lemma 4.1 this becomes S F ( s ) = − ω F e − sgn( t ) πis (cid:0) O ( e − c | t | ) (cid:1) (5.14)since ω F = ±
1. Similarly, for virtual γ -factors of Hecke type we have, thanks to our assumptionon µ , that S γ ( s ) = − ie πiµ e πis + ie − πiµ e − πis = − η e − sgn( t ) πis (cid:0) O ( e − c | t | ) (cid:1) (5.15)with some c > η = ie πiµ ∈ {− , } . (5.16)The result follows in this case from (5.14),(5.15) and (5.16) with η = ω F /η ∈ {− , } .For virtual γ -factors of Maass type we have S γ ( s ) = − ( − ε e − sgn( t ) πis (cid:0) O ( e − c | t | ) (cid:1) (5.17)with some c >
0. Since in this case ω F = ( − ε by Proposition 5.1, the result follows with η = 1 from (5.14) and (5.17). (cid:3) Finally, with a convenient abuse of notation, we rewrite (2.7) as S F ( s ) = N X j =0 a j e iπω j s (5.18)with N ≥ , − ω < ω < · · · < ω N = 1 , ω j = − ω N − j , a j = 0 (5.19)and, by (5.14), a = a N = − ω F . (5.20) Lemma 5.4.
Under the same hypotheses of Lemma . , and with the notation in (5.18) andin Lemma . , we have that if R ( s ) = η then N ≥ and ω N − > . Proof.
We start with the case of Hecke type virtual γ -factors. From (5.15) and (5.16) wehave S γ ( s ) = − η (cid:0) e πis + e − πis (cid:1) = − η cos( πs ) . (5.21)If N = 1, from (5.20) we get S F ( s ) = − ω F (cid:0) e πis + e − πis (cid:1) = − ω F cos( πs ) (5.22)hence R ( s ) = η thanks to Lemma 5.3, (5.16) and (5.21). If N = 2 we have S F ( s ) = − ω F e − πis + a e πiω s − ω F e πis and ω = − ω by (5.19), so ω = 0. Thus, again by (5.19), S F ( s ) = − ω F cos( πs ) + a , a = 0 . (5.23)Applying twice the functional equation in Proposition 5.1 we get R ( s ) R (1 − s ) = 1 . (5.24)Thus, from (5.24),(5.16) and (5.21) we deduce that S F ( s ) S F (1 − s ) = S γ ( s ) S γ (1 − s ) = 4 cos( πs ) cos( π (1 − s )) = − ( πs ) . Hence if S F ( x ) = 0 then cos( πx ) = 0, which contradicts (5.23). Thus we cannot have N = 2,and the result follows in this case.For virtual γ -factors of Maass type we have S γ ( s ) = 4 sin (cid:0) π ( s + ε + iκ (cid:1) sin (cid:0) π ( s + ε − iκ (cid:1) = − − ε cos( πs ) + a ( γ ) (5.25)with a certain a ( γ ). If N = 1, from (5.24),(5.22) and the first equation in (5.25) we get16 sin (cid:0) π ( s + ε + iκ (cid:1) sin (cid:0) π ( s + ε − iκ (cid:1) sin (cid:0) π ( 1 − s + ε + iκ (cid:1) sin (cid:0) π ( 1 − s + ε − iκ (cid:1) = 4 cos ( πs ) , impossible since zeros on the two sides do not match. Thus we cannot have N = 1. If N = 2,then by (5.24),(5.23) and the second expression for S γ ( s ) in (5.25) we have (cid:0) − ω F cos( πs ) + a (cid:1)(cid:0) − ω F cos( π (1 − s )) + a (cid:1) = (cid:0) − − ε cos( πs ) + a ( γ ) (cid:1) ( − − ε cos( π (1 − s )) + a ( γ ) (cid:1) , which gives a = a ( γ ) and hence a = ± a ( γ ). Recalling that ω F = ( − ε and η = 1, seeProposition 5.1 and Lemma 5.3 (in this case we have χ F ≤ γ -factor is ofMaass type), if a = a ( γ ) we have S F ( s ) = ηS γ ( s ) by (5.23) and (5.25). If a = − a ( γ ), thenfrom (5.23) and (5.25) we have S γ ( s ) = − S F (1 − s ) and hence Corollary 5.1 yelds S F (1 − s ) γ ( s ) F ( s ) = − ω F S F ( s ) γ (1 − s ) F (1 − s ) . Dividing this by the functional equation (2.1) of F , denoting by γ F ( s ) the γ -factor of F andrecalling Lemma 4.1 we obtain S F (1 − s ) γ ( s ) γ F ( s ) = − S F ( s ) γ (1 − s ) γ F (1 − s ) . Thus the function f ( s ) = S F (1 − s ) γ ( s ) γ F (1 − s ) satisfies f ( s ) = − f (1 − s ), and in particular f (1 /
2) = 0. Therefore S F (1 / γ (1 / γ F (1 /
2) = 0 , and since γ -factors do not vanish this implies 0 = S F (1 /
2) = a , which contradicts (5.23).Hence for N = 2 we have R ( s ) = η and the result follows. (cid:3) Period functions
For a function F as in Theorem 1 . z in the upper half-plane H we define f ( z ) = ∞ X n =1 a ( n ) n λ e ( nz ) , where λ = ( µ if χ F > iκ if χ F ≤
0, (6.1) µ, κ are as in (5.1) and χ F is defined by (1.1). Clearly, f ( z ) is holomorphic on H . We shallrepeatedly use the following simple, and essentially well known, lemma. Lemma 6.1.
For every F ∈ S ♯ with positive degree and λ ∈ C the function f ( z ) cannot becontinued to an entire function. Proof.
From the convergence properties of the Dirichlet series of F we have that a ( n ) ≪ n ε ,hence the power series g ( z ) = P ∞ n =1 a ( n ) n λ z n is convergent for | z | <
1. If f ( z ) is entire, then g ( z ) is also entire since clearly it is bounded around z = 0. Therefore g ( z ) is convergent forsome | z | >
1, thus a ( n ) n λ ≪ | z | − n and hence a ( n ) ≪ n − A for every A >
0. As a consequencethe Dirichlet series of F is everywhere convergent, a contradiction since its Lindel¨of function µ F ( σ ) is positive for σ < (cid:3) In the next two propositions we study the function f ( z ) distinguishing the cases χ F > χ F ≤
0, corresponding to associated virtual γ -factors of Hecke and Maass type, since thearguments are somewhat different. In the first case we also use the properties of f ( z ) to showthat the hypothesis on the parameter µ in Lemmas 5.3 and 5.4 holds true. However, for easeof reference we adopt the following uniform notation.Let R ( s ) and η be as in Lemma 5.3, γ being the virtual γ -factor associated with F . Writing0 < c < ℜ ( λ ) = 0 and 0 < c < ℜ ( λ ) if ℜ ( λ ) >
0, and c = 1 + 2 ℜ ( λ ) − c , (6.2)we define the following functions.(a) P ( z ) ≡ λ = µ , i.e. γ is of Hecke type, while P ( z ) = i ε +1 cos( πλ )2 π λ +3 / πi Z ( c ) Γ (cid:0) − ε + s (cid:1) Γ (cid:0) − ε − s + 2 λ (cid:1) γ ( s − λ ) F ( s − λ ) z s − − λ d s (6.3)if λ = iκ , i.e. γ is of Maass type; clearly, the integral is absolutely convergent for | arg( z ) | < π and P ( z ) is holomorphic there.(b) For every λ as above Q η ( z ) = 12 πi Z ( c ) (cid:0) R ( s − λ ) − η )Γ( s ) γ (1 − s + λ ) γ ( s − λ ) F (1 − s + λ )( − πiz ) − s d s ; (6.4)by Lemma 5.3 the integral is certainly absolutely convergent for z ∈ H and Q η ( z ) is holomorphicthere. (c) For every λ as above L ( z ) = res s =1+ λ Γ( s ) F ( s − λ )( − πiz ) − s ; (6.5)clearly, L ( z ) ≡ F is entire and otherwise is holomorphic for | arg( z ) | < π , since it is a linearcombinations of terms of type z a and z a log k z with a ∈ C and k ∈ N . Finally, for z ∈ H wewrite ψ ( z ) = f ( z ) − z − λ − f ( − /z ) . (6.6) Proposition 6.1.
Let F be as in Theorem . with χ F > . Then µ + 1 / is a positiveinteger and ψ ( z ) = ηP ( z ) + L ( z ) + ω F Q η ( z ) , where P ( z ) , L ( z ) and Q η ( z ) are holomorphic for | arg( z ) | < π . Proof.
For future reference we take the first steps of the proof keeping χ F general and usingthe above uniform notation. Let z = x + iy ∈ H . Since F ( s − λ ) is absolutely convergent for σ > ℜ ( λ ), by Mellin’s transform we have f ( iy ) = ∞ X n =1 a ( n ) n λ e − πny = 12 πi Z ( c ) (2 π ) − s Γ( s ) F ( s − λ ) y − s d s, (6.7)where c > ℜ ( λ ) + 1. Recalling (6.5) and that F has at most a pole at s = 1, we shift theintegration line to σ = c as in (6.2), thus getting f ( iy ) = 12 πi Z ( c ) (2 π ) − s Γ( s ) F ( s − λ ) y − s d s + L ( iy ) . Hence by Corollary 5.1 we obtain f ( iy ) = ω F πi Z ( c ) (2 π ) − s Γ( s ) R ( s − λ ) γ (1 − s + λ ) γ ( s − λ ) F (1 − s + λ ) y − s d s + L ( iy ) . (6.8)Now we recall that if χ F > λ = µ > γ -factor is γ ( s ) =(2 π ) − s Γ( s + µ ); in what follows we restrict ourselves to this case. Thus, for χ F >
0, by thechange of variable s − s + µ we see that (6.8) becomes f ( iy ) = ω F (2 π ) − µ y − µ − πi Z (1 − c + µ ) R (1 − s ) γ ( s ) F ( s ) y s d s + L ( iy ) . (6.9)But 1 − c + µ > − c + µ = 1 + δ with some δ >
0; for later use, wemay assume that δ is sufficiently small. Moreover, thanks to (5.24) we rewrite (6.9) as f ( iy ) = ω F (2 π ) − µ y − µ − πi Z (1+ δ ) S γ ( s ) S F ( s ) γ ( s ) F ( s ) y s d s + L ( iy ) . Hence, expanding F ( s ), recalling that S γ ( s ) = 2 sin( π ( s + µ )) by definition and applying thereflection formula to γ ( s ), we finally obtain that f ( iy ) = ω F (2 π ) − µ y − µ − ∞ X n =1 a ( n ) 12 πi Z (1+ δ ) S F ( s )Γ(1 − s − µ ) (cid:16) πny (cid:17) − s d s + L ( iy ) . (6.10)From (5.18)-(5.20) we have that S F ( s ) = − ω F cos( πs ) + N − X j =1 a j e iπω j s with | ω j | < ≤ j ≤ N −
1. Thus, as | t | → ∞ S F ( s ) + 12 ω F cos( πs ) ≪ e − π (1+ ρ ) | t | (6.11)for some 0 < ρ < /
2. Hence we rewrite (6.10) as f ( iy ) = −
12 (2 π ) − µ y − µ − ∞ X n =1 a ( n ) J (cid:16) πny (cid:17) + H ( y ) + L ( iy ) (6.12)where, introducing a new variable w for later use, J ( w ) = 12 πi Z (1+ δ ) πs )Γ(1 − s − µ ) w − s d s and H ( y ) = ω F (2 π ) − µ y − µ − ∞ X n =1 a ( n ) 12 πi Z (1+ δ ) n S F ( s ) + 12 ω F cos( πs ) o − s − µ ) (cid:16) πny (cid:17) − s d s. Now we consider y as a complex variable. From (6.11) and Stirling’s formula we see that thesum of integrals defining H ( y ) is holomorphic for | arg( y ) | < π/ πρ . Moreover, by Stirling’sformula the integral defining J (2 πn/y ) is holomorphic for | arg( y ) | < π/
2, and clearly J (2 πn/y ) ≪ y n − − δ . Thus the series on the right hand side of (6.12) is also holomorphic for | arg( y ) | < π/ J ( w ). To this end we shift the line of integration to −∞ and, recalling that δ > s = − ℓ + 1 / ℓ ≥
0. Thus we obtain that J ( w ) = − w − / π ∞ X ℓ =0 ( − w ) ℓ Γ( ℓ + 1 / − µ ) = − w − / π E / − µ ( w ) , (6.13)say. Formula (6.13) holds since by Stirling’s formula the integral tends to 0 as the integrationline is shifted to −∞ , and the series in (6.13) is everywhere convergent; in particular, E / − µ ( w )is an entire function. Actually, E / − µ ( w ) is a special case of a two-parametric Mittag-Lefflerfunction , see Chapter 4 of Gorenflo-Kilbas-Mainardi-Rogosin [3]. More precisely we have that E β ( w ) = E ,β ( − w ) , where β = 1 / − µ and E ,β ( w ) is the function defined by (4.1.1) of [3]. One easily checks(see also (4.3.2) of [3] with m = n = 1) that E β ( w ) satisfies the first order linear differentialequation E ′ β ( w ) = − (cid:16) β − w (cid:17) E β ( w ) + 1 w Γ( β − . Writing w = u + iv , we solve such a differential equation by the standard method, choosingan initial point w = 0 of the form u + iv and then letting u → −∞ . Thus we obtain for w ∈ C \ [0 , + ∞ ) that E β ( w ) = κ e − w w − β + e − w w − β Γ( β − Z w −∞ e ξ ξ β − d ξ (6.14)with some constant κ , where the integration is along the horizontal half-line ξ = β + iv , −∞ < β ≤ u , and the integral is absolutely convergent. Formula (6.14) follows also from(4.3.6) of [3] with z = − w , z = − w and n = 1, after the change of variable τ = − ξ inside theintegral. Indeed, the constant term in (4.3.6) is convergent as u → −∞ thanks to (4.4.16) inTheorem 4.3 of [3]. In view of (6.1), from (6.12)-(6.14) with w = 2 πn/y we get f ( iy ) = κ ( iy ) − µ − f ( i/y ) + y − µ − Γ( − / − µ ) e f ( y ) + H ( y ) + L ( iy ) , (6.15)where κ = κ i µ +1 (6.16)and e f ( y ) = ∞ X n =1 a ( n ) n µ e − πn/y Z πn/y −∞ e ξ ξ − µ − / d ξ. (6.17)Note that the integral in (6.17) is ≪ e πn ℜ ( y ) / | y | n − µ − / uniformly for 2 πn/y in any compact subset of C \ [0 , + ∞ ), therefore the series in (6.17) isabsolutely convergent for the same values of y and hence e f ( y ) is holomorphic for y ∈ C \ [0 , + ∞ ).But, since arg( iy ) = π/ y ), every other term in (6.15) is holomorphic at least for | arg( y ) | < π/
2, and H ( y ) + L ( iy ) is holomorphic for | arg( y ) | < π/ πρ . Hence, coming backto the variable z = x + iy , we rewrite (6.15) as f ( z ) = κ z − µ − f ( − /z ) + e ψ ( z ) , (6.18)where e ψ ( z ) is holomorphic for − πρ < arg( z ) < π + πρ .Now we are ready to conclude the proof. Applying again (6.18) to f ( − /z ) we obtain that f ( z ) = κ f ( z ) + e H ( z )with a certain function e H ( z ) holomorphic for − πρ < arg( z ) < π + πρ . If κ = ±
1, then f ( z )is also holomorphic for − πρ < arg( z ) < π + πρ . But f ( z ) is periodic of period 1 hence it isentire, a contradiction by Lemma 6.1; thus κ = ± . (6.19)Applying (6.18) with z + 1 in place of z and recalling the periodicity of f ( z ) we get1 κ e ψ ( z + 1) = 1 κ f ( z ) − ( z + 1) − µ − f (cid:16) − z + 1 (cid:17) . (6.20)Subtracting (6.20) from (6.18), using the 1-periodicity in the resulting terms f ( − / ( z + 1)) and f ( − /z ), and then applying again (6.18) with z/ ( z + 1) in place of z we finally obtain that e ψ ( z ) − κ e ψ ( z + 1) = (cid:16) − κ (cid:17) f ( z ) − κ z − µ − f ( − /z ) + ( z + 1) − µ − f (cid:16) − z + 1 (cid:17) = (cid:16) − κ (cid:17) f ( z ) + ( z + 1) − µ − e ψ (cid:16) zz + 1 (cid:17) . (6.21)Suppose now that κ = 1. Then (6.21) gives an expression of f ( z ) in terms of e ψ ( z ), e ψ ( z + 1)and e ψ ( z/ ( z + 1)). But, in particular, e ψ ( z ) is holomorphic for | arg( z ) | < πρ , and clearly | arg( z + 1) | , | arg( z/ ( z + 1)) | < πρ for z in that region. Therefore, such an expression showsthat f ( z ) is holomorphic for | arg( z ) | < πρ , thus it is entire by 1-periodicity, a contradiction byLemma 6.1. Consequently by (6.16) and (6.19) we have1 = κ = κ i µ +1 = ± i µ +1 and hence µ + 1 / e ψ ( z ) = ψ ( z ) in view of (6.6), thus ψ ( z ) is holomorphicfor − πρ < arg( z ) < π + πρ , and (6.21) becomes the three-term functional equation ψ ( z ) = ψ ( z + 1) + ( z + 1) − µ − ψ (cid:0) zz + 1 (cid:1) . (6.22)To prove the second assertion we first recall that γ ( s ) = (2 π ) − s Γ( s + µ ), c = 1 − c + 2 µ > η is as in Lemma 5.3. Starting with (6.8), adding and subtracting η to R ( s − µ ), separatingthe two integrals involving η and R ( s − µ ) − η and finally making the substitution s − s +2 µ into the first one, in view of (6.7) we have f ( iy ) = ηω F (2 π ) − µ y − µ − πi Z ( c ) Γ( s ) F ( s − µ ) (cid:16) πy (cid:17) − s + µ d s + L ( iy ) + ω F Q η ( iy )= ( iy ) − µ − f ( i/y ) + L ( iy ) + ω F Q η ( iy ) . Thus, recalling (6.6) and the regularity properties of ψ ( z ) and L ( z ), by analytic continuationwe obtain ψ ( z ) = L ( z ) + ω F Q η ( z ) (6.23)for − πρ < arg( z ) < π + πρ . In order to extend the range of holomorphy in (6.23) we exploit(6.22). Indeed, by elementary geometry, we have | arg (cid:0) zz + 1 (cid:1) | , | arg( z + 1) | < | arg( z ) | for | arg( z ) | < π . More precisely, given R, ε >
0, if z ∈ C ( R, ε ) = {| z | ≤ R : | arg( z ) | ≤ π − ε } then | arg (cid:0) zz + 1 (cid:1) | , | arg( z + 1) | ≤ | arg( z ) | − δ ( R, ε )for some δ ( R, ε ) >
0. Thus (6.22) gives step-by-step analytic continuation of ψ ( z ) to C ( R, ε ),and hence to | arg( z ) | < π since R and ε are arbitrary. The result follows now from (6.23) since P ≡ L ( z ) is holomorphic for | arg( z ) | < π . (cid:3) Proposition 6.2.
Let F be as in Theorem . with χ F ≤ . Then ψ ( z ) satisfies the sameproperties as in Proposition . . Proof.
For simplicity we keep the uniform notation described at the beginning of this section,but later on in the proof we use the fact that the virtual γ -factor γ ( s ) is of Maass type. Wefollow the steps in the proof of Proposition 6.1 till equation (6.8); then, by Lemma 5.3, in viewof (6.4) we have f ( iy ) = η ω F πi Z ( c ) (2 π ) − s Γ( s ) γ (1 − s + λ ) γ ( s − λ ) F (1 − s + λ ) y − s d s + L ( iy )+ ω F πi Z ( c ) (2 π ) − s Γ( s ) (cid:0) R ( s − λ ) − η (cid:1) γ (1 − s + λ ) γ ( s − λ ) F (1 − s + λ ) y − s d s = ηP ( iy ) + L ( iy ) + ω F Q η ( iy ) , (6.24)say. In P ( iy ) we make the substitution 1 − s + λ s − λ . Thus, recalling (6.2) and writing T ( s ) = (2 π ) s − − λ Γ(1 − s + 2 λ ) γ ( s − λ )Γ( s ) γ (1 − s + λ ) , ω ( λ ) = ω F e i π (1+2 λ ) , (6.25) we obtain that P ( iy ) = ω F πi Z ( c ) (2 π ) s − − λ Γ(1 − s + 2 λ ) γ ( s − λ ) γ (1 − s + λ ) F ( s − λ ) y s − − λ d s = ω ( λ ) ( iy ) − − λ πi Z ( c ) (2 π ) − s Γ( s ) F ( s − λ ) (cid:0) y (cid:1) − s T ( s )d s. (6.26)Now, thanks to (6.7) and (6.25), recalling that c − ℜ ( λ ) > P ( iy ) = ( iy ) − − λ πi Z ( c ) (2 π ) − s Γ( s ) F ( s − λ ) (cid:0) y (cid:1) − s d s + ( iy ) − − λ πi Z ( c ) (2 π ) − s Γ( s ) F ( s − λ ) (cid:0) y (cid:1) − s (cid:0) ω ( λ ) T ( s ) − (cid:1) d s = ( iy ) − λ − f (cid:0) − / ( iy ) (cid:1) + P ( iy ) , (6.27)say. Since η = 1 in this case thanks to Lemma 5.3, from (6.6),(6.24) and (6.27) we have byanalytic continuation that ψ ( z ) = ηP ( z ) + L ( z ) + ω F Q η ( z ) . (6.28)Hence Proposition 6.2 follows from (6.28) once we show that P ( z ) can be written as in (6.3)and Q η ( z ) is holomorphic for | arg( z ) | < π .To this end we first rewrite P ( iy ) as P ( iy ) = ( iy ) − − λ πi Z ( c ) γ ( s − λ ) F ( s − λ ) (cid:0) πy (cid:1) − s × (cid:16) ω ( λ )(2 π ) s − − λ Γ(1 − s + 2 λ ) γ (1 − s + λ ) − Γ( s ) γ ( s − λ ) (cid:17) d s. (6.29)But, in view of (5.1), using the duplication and reflection formulae for the Γ function and thefact that ε ∈ { , } we haveΓ(1 − s + 2 λ ) γ (1 − s + λ ) = Γ(1 − s + 2 λ ) π s − λ − Γ (cid:0) − s + ε +2 λ (cid:1) Γ (cid:0) − s + ε (cid:1) = π / λ − s − s +2 λ Γ (cid:0) − s − ε +2 λ (cid:1) Γ (cid:0) − s + ε (cid:1) = π / λ − s − s +2 λ Γ (cid:0) − s − ε +2 λ (cid:1) Γ (cid:0) s − ε (cid:1) Γ (cid:0) − s + ε (cid:1) Γ (cid:0) s − ε (cid:1) = π − / λ − s − s +2 λ Γ (cid:0) − s − ε + 2 λ (cid:1) Γ (cid:0) s − ε (cid:1) sin (cid:0) π (cid:0) − s + ε (cid:1)(cid:1) and Γ( s ) γ ( s − λ ) = π s − λ − / s − Γ (cid:0) s (cid:1) Γ (cid:0) s +12 (cid:1) Γ (cid:0) s + ε (cid:1) Γ (cid:0) s + ε − λ (cid:1) = π s − λ − / s − Γ (cid:0) s +1 − ε (cid:1) Γ (cid:0) s + ε − λ (cid:1) = π s − λ − / s − Γ (cid:0) s +1 − ε (cid:1) Γ (cid:0) − s − ε +2 λ (cid:1) Γ (cid:0) s + ε − λ (cid:1) Γ (cid:0) − s + ε − λ (cid:1) = π s − λ − / s − Γ (cid:0) s − ε (cid:1) Γ (cid:0) − s − ε + 2 λ (cid:1) sin (cid:0) π (cid:0) s + ε − λ (cid:1)(cid:1) . Consequently ω ( λ )(2 π ) s − − λ Γ(1 − s + 2 λ ) γ (1 − s + λ ) − Γ( s ) γ ( s − λ ) = (2 π ) s π λ +3 / Γ (cid:0) s − ε (cid:1) Γ (cid:0) − s − ε + 2 λ (cid:1) × (cid:0) ω ( λ ) sin (cid:0) π (cid:0) − s + ε (cid:1) − sin (cid:0) π (cid:0) s + ε − λ (cid:1)(cid:1) , (6.30) and clearly ω ( λ ) sin (cid:0) π (cid:0) − s + ε (cid:1) − sin (cid:0) π (cid:0) s + ε − λ (cid:1) = 12 i (cid:16) − ω F e i π (2 λ − s + ε ) − ω F e i π (2 λ + s − ε ) − e i π ( s + ε − λ ) + e − i π ( s + ε − λ ) (cid:17) . Recalling that in this case ω F = ( − ε , see Proposition 5.1, we have − ω F e i π (2 λ − s + ε ) + e − i π ( s + ε − λ ) = 0 , and hence ω ( λ ) sin (cid:0) π (cid:0) − s + ε (cid:1) − sin (cid:0) π (cid:0) s + ε − λ (cid:1) = ie i π s e i π ε cos( πλ ) . (6.31)Gathering (6.29),(6.30) and (6.31) we see that P ( iy ) has the required form (6.3).Finally, from (6.4) and Lemma 5.3 we have that Q η ( z ) is holomorphic for − δ < arg( z ) < π + δ with some δ >
0, hence the same holds for ψ ( z ) by (6.28). But, as in the proof of Proposition6.1, from the 1-periodicity of f ( z ) we deduce that ψ ( z ) satisfies the three-term functionalequation (6.22), thus the argument at the end of the proof of Proposition 6.1 shows that ψ ( z )is holomorphic for | arg( z ) | < π also in this case. Proposition 6.2 follows now from (6.28). (cid:3) Remark 6.1.
The computations in the proof of Proposition 6.2 are similar to those inLewis-Zagier [19], pages 204–205. Apparently there are some slight differences, unimportantin our case, in the final formulae. The function ψ ( z ) in (6.6) is a period function in the senseof Lewis-Zagier [19]. The fact that ψ ( z ) in (6.28) is holomorphic for | arg( z ) | < π follows fromthe results in Section 4 of Chapter 3 of [19]; the above argument gives an independent directproof, which works also for certain functional equations more general than (6.22). (cid:3) Conclusion of the proof
Now we are ready to conclude the proof of Theorem 1.1. Our aim is to show that the function R ( s ) in Corollary 5.1 is identically equal to η . Indeed, if this is the case then S F ( s ) = ηS γ ( s ),hence Corollary 5.1 implies that F satisfies the functions equation γ ( s ) F ( s ) = ωγ (1 − s ) F (1 − s ) with ω = ω F η. We are therefore in the situation of Lemma 5.1. More precisely, by (5.2) and Proposition 5.1, if χ F > χ F = 0 then κ = 0 and hence by Satz 2 of Maass [20] we have that F ( s ) = ζ ( s ) . Theorem 1.1 thus follows.Note that computing the invariant ω F by means of (2.3), starting with the above γ -factor and ω -datum, we obtain that ω F = ω F η , thus η = 1 in all cases.Suppose, by contradiction, that R ( s ) η . Let G ( z, s ) = Γ( s ) γ (1 − s + λ ) γ ( s − λ ) F (1 − s + λ )( − πiz ) − s . (7.1)Thanks to Lemma 5.4, (5.18) and (5.19), we rewrite the function Q η ( z ) in (6.4) as Q η ( z ) = 12 πi (cid:16) Z c c − i ∞ + Z c + i ∞ c (cid:17)(cid:0) R ( s − λ ) − η ) G ( z, s )d s = Q − η ( z ) + Q + η ( z )= Q + η ( z ) + N − X j =0 a j πi Z c c − i ∞ e iπω j ( s − λ ) S γ ( s − λ ) G ( z, s )d s + a N − πi Z c c − i ∞ e iπω N − ( s − λ ) S γ ( s − λ ) G ( z, s )d s + 12 πi Z c c − i ∞ (cid:16) a N e iπ ( s − λ ) S γ ( s − λ ) − η (cid:17) G ( z, s )d s = Q + η ( z ) + A ( z ) + B ( z ) + C ( z ) , (7.2) say. In view of Lemma 5.3, as | t | → ∞ the integrand in Q + η ( z ) is certainly ≪ ( | t | + 1) c e (arg( z ) − π ) | t | for some c > Q + η ( z ) is holomorphic for | arg( z ) | < π . (7.3)Recalling (5.21), for 0 ≤ j ≤ N − t → −∞ we have e iπω j ( s − λ ) S γ ( s − λ ) G ( z, s ) ≪ ( | t | + 1) c e −| t | (arg( z )+ π (1 − ω j )) for some c >
0, hence by (5.19) A ( z ) is holomorphic for − π min(1 − ω N − , < arg( z ) < π . (7.4)Moreover, recalling the value of η in Lemma 5.3 and equations (5.15),(5.16),(5.17) and (5.20),we observe that the main terms of a N e iπ ( s − λ ) − ηS γ ( s − λ ) cancel, hence as t → −∞ a N e iπ ( s − λ ) S γ ( s − λ ) − η ≪ e − π | t | . Thus a computation as above shows that C ( z ) is holomorphic for | arg( z ) | < π . (7.5)Finally, to treat B ( z ) we observe from (5.21) and (5.25) that as t → −∞ S γ ( s − λ ) = ρ e − iπs + O ( e − π | t | )with a constant ρ ∈ C \ { } , hence B ( z ) = ρ πi Z c c − i ∞ e iπs ( ω N − − G ( z, s )d s + D ( z ) (7.6)with a constant ρ ∈ C \{ } and a function D ( z ) holomorphic for | arg( z ) | < π . But as t → + ∞ the integrand in (7.6) is ≪ ( | t | + 1) c e −| t | ( πω N − − arg( z )) for some c >
0. Thus, recalling from Lemma 5.4 that ω N − >
0, we have that ρ πi Z c + i ∞ c e iπs ( ω N − − G ( z, s )d s is holomorphic for − π < arg( z ) < πω N − . (7.7)Therefore, gathering (7.1)-(7.7), from Propositions 6.1 and 6.2 we deduce that the integral I ( z ) = 12 πi Z ( c ) Γ( s ) γ (1 − s + λ ) γ ( s − λ ) F (1 − s + λ )( − πie iπ (1 − ω N − ) z ) − s d s (7.8)represents a holomorphic function for − π min(1 − ω N − , < arg( z ) < πω N − .Thanks to (5.7), Proposition 5.1, (2.10),(2.1) and Lemma 4.1, recalling that γ F ( s ) denotesthe γ -factor of F we have γ (1 − s + λ ) γ ( s − λ ) = γ F (1 − s + λ ) γ F ( s − λ ) S γ ( s − λ ) S F ( s − λ )= η − γ F (1 − s + λ ) γ F ( s − λ ) + γ F (1 − s + λ ) γ F ( s − λ ) (cid:16) S γ ( s − λ ) S F ( s − λ ) − η − (cid:17) = η − γ F (1 − s + λ ) γ F ( s − λ ) + O ( e − δ | t | ) with some δ > γ (1 − s + λ ) γ ( s − λ ) by η − γ F (1 − s + λ ) γ F ( s − λ ) in (7.8) adds anextra term which is holomorphic at least for − π min(1 − ω N − + δ, < arg( z ) < πω N − . Thus,writing w = w ( z ) := − ie iπ (1 − ω N − ) z, the integral J ( z ) = 12 πi Z ( c ) Γ( s ) γ F (1 − s + λ ) γ F ( s − λ ) F (1 − s + λ )(2 πw ) − s d s represents a holomorphic function for − π min(1 − ω N − , − ω N − + δ, < arg( z ) < πω N − . (7.9)But by (2.1) J ( z ) = ω F πi Z ( c ) Γ( s ) F ( s − λ )(2 πw ) − s d s. Therefore, recalling the choice of c in (6.2) and shifting the integration line to σ = c > ℜ ( λ ),by Mellin’s transform we get J ( z ) = ω F πi Z ( c ) Γ( s ) F ( s − λ )(2 πw ) − s d s − ω F res s =1+ λ Γ( s ) F ( s − λ )(2 πw ) − s = ω F f (cid:0) e iπ (1 − ω N − ) z (cid:1) − ω F res s =1+ λ Γ( s ) F ( s − λ )(2 πw ) − s . (7.10)Thus f (cid:0) e iπ (1 − ω N − ) z (cid:1) is holomorphic in the region (7.9), hence changing variable we get that f ( z ) is holomorphic for − π min( ω N − − ω N − , δ, ω N − ) < arg( z ) < π . Recalling that ω N − > f ( z ) is holomorphic for − δ < arg( z ) < π for some δ > f ( z ) is 1-periodic, hence it is entire, a contradiction by Lemma 6.1. Theorem 1.1 is nowproved. (cid:3) References [1] B.C.Berndt, M.I.Knopp -
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