arXiv:1606.05726v3 [math.NT] 1 Oct 2020 h ok fLgra 2,2,2](n loBro 1,1,12 11, [10, Burnol also wa (and study 27] present 26, The [25, dealing Lagarias by introduction. of relation the works the the in in explain function an We c zeta studying type. of mann [45] special terms paper a in preliminary of class systems Selberg the the in of results subclass general wide a in functions lse fzt-iefntos seily naaou of analogue an Especially, functions. zeta-like of classes ebr ls sotncle h rn imn yohss( Hypothesis Riemann Grand the called often is class Selberg function a tnsfrtecmlxcnuae ti aiycnre t confirmed easily is It conjugate. complex the for stands bar ya niefunction entire an by ytefntoa equations functional the by o nntl ayetr functions entire many infinitely for sa niefnto aigra-auso h rtcllin critical the of function on zeros real-values nontrivial taking with function entire an is decomposition F s tfrteGadReanHptei hogotti pape this throughout Hypothesis Riemann Grand the for it use class ♯ h imn xi-function Riemann The h rsn ae ist sals e qiaetconditi equivalent new a establish to aims paper present The h imn yohss(H set htalnnrva zer nontrivial all that asserts (RH) Hypothesis Riemann The 2010 h datg ftedcmoiin(.)sad ntetheor the on stands (1.1) decomposition the of advantage The 1 = h brvainGHi fe sdfrteGnrlzdRiema Generalized the for used often is GRH abbreviation The fetr ucin.W eoeby denote We functions. entire of − ahmtc ujc Classification. Subject Mathematics F nivrepolmfrcnnclsseso pca type. special a equi of the systems establish canonical To for problem Hamiltonian. inverse an a called determined is function system valued canonical A equations. differential of ssfor esis Abstract. ξ ζ . ( ( s s i nteciia ie oigti,w tr ihconside with start we this, Noting line. critical the on lie ) i nteciia line critical the on lie ) L fntosi iesbls fteSlegcasi em of terms in class Selberg the of subclass wide a in -functions L eetbihanweuvln odto o h rn Rieman Grand the for condition equivalent new a establish We FNTOSI H EBR CLASS SELBERG THE IN -FUNCTIONS E AITNASAIIGFROM ARISING HAMILTONIANS where , E ζ ( ξ s | ξ ( E .TeeoeR seuvln htalzrso h entire the of zeros all that equivalent is RH Therefore ). z ( ξ ξ s = ) ( ♯ = ) s ( i z 1 2 = ) ) = − | AAOH SUZUKI MASATOSHI 1. 1 2 < ξ √ E iz ξ s ( (1 ℜ s − ( | Introduction = E 14,3A5 11M26. 34A55, 11M41, s + ) ( 1, s − − = ( HB E z 1 = ) F ) ξ 1) s 2 1 ds | d and ) + ♯ 1 ( h e faletr ucin satisfying functions entire all of set the π ( E ξ z F − o all for ( = ) / ( s/ s where , z ,adi a engnrlzdt wider to generalized been has it and 2, ) 2 + ) ξ Γ F (
s s (¯ E z = ) z 2 s = r. o netr function entire an for ) ♯ ∈ ( F 2 1 z ζ C ξ yara ymti matrix- symmetric real a by )(1.1) )) ( − uhta h eo coincide zeros the that such e sa niefnto satisfying function entire an is ♯ + nHptei nltrtr u we but literature in Hypothesis nn s ( aetcniin euse we condition, valent ) s , iz Hfor RH es rbe fcanonical of problem verse n hs(.)as holds also (1.1) thus and ) nnclssesb using by systems anonical a 11 od for holds (1.1) hat GRH) 3 swl s[45]). as well as 13] , . ihtecs fteRie- the of case the with fthe of y so h imn zeta- Riemann the of os anysiuae by stimulated mainly s aoia systems canonical nfrGHfor GRH for on 1 L . fntosi the in -functions Hermite–Biehler Hypoth- n iga additive an ring F n the and (1.2) (1.3) L - 2 M. SUZUKI where C = z = u + iv u, v R,v> 0 . The Hermite–Biehler class HB consists of all + { | ∈ } E HB having no real zeros. If E HB, the two entire functions ∈ ∈ 1 i A(z) := (E(z)+ E♯(z)) and B(z) := (E(z) E♯(z)) (1.4) 2 2 − have only real zeros. Further, all (real) zeros of A and B are simple if E HB ([4]). Therefore, the existence of E HB satisfying (1.1) implies that RH holds together∈ with the Simplicity Conjecture (SC)∈ which asserts that all nontrivial zeros of ζ(s) are simple. Conversely, there exists E HB satisfying (1.1) if we assume that RH and SC hold. In ∈ fact, Eξ of (1.2) belongs to HB under RH and SC ([26]). The above discussion suggests the following strategy to the proof of RH and SC: first, find an entire function E satisfying (1.1); second, prove that E belongs to HB. Then these two conditions conclude RH and SC. More simply, we may start from the second step by using Eξ of (1.2). The obvious difficulty with this strategy is in the second step, because the reason why an entire function E satisfying (1.1) (or Eξ) belongs to HB is not known other than RH and SC, so we do not know how to prove E HB without RH and SC. In this paper, we propose a strategy to prove E HB with∈ the help of de Branges’ theory of canonical systems as with Lagarias [25, 26,∈ 27] in which the applicability of the theory of de Branges to the study of the zeros of L-functions is suggested. To start with, we explain how canonical systems generate functions of the class HB. A 2 2 real symmetric matrix-valued function H defined on I = [t ,t ) ( < t < × 0 1 −∞ 0 t1 ) is called a Hamiltonian if H(t) is positive semidefinite for almost every t I, H ≤ ∞0 on any subset of I with positive Lebesgue measure, and H is locally integrable∈ on 6≡I with respect to the Lebesgue measure. The first-order system d A(t, z) 0 1 A(t, z) = z − H(t) , z C (1.5) − dt B(t, z) 1 0 B(t, z) ∈ associated with a Hamiltonian H on I is called a canonical system on I (See [45, Sec- tion 1] for a difference with the usual definition). For a solution t[A(t, z),B(t, z)] of a canonical system, we define A(t, z)B(t, w) A(t, w)B(t, z) J(t; z, w) := − . (1.6) π(w z¯) − Then, E(t, z) := A(t, z) iB(t, z) is an entire function of HB for every t I if we suppose that lim J(t;−z, z) = 0 for every z C , unless det H(t) = 0 for∈ almost t→t1 ∈ + every t I (see [45, Proposition 2.4] for details). Conversely, for E HB, there exists ∈ ∈ a Hamiltonian H defined on a (possibly unbounded) interval I = [t0,t1) such that for a unique solution t[A(t, z),B(t, z)] of (1.5) satisfying E(z) = A(t , z) iB(t , z), the 0 − 0 function E(t, z)= A(t, z) iB(t, z) belongs to HB for every t I, and J(t; z, w) defined by (1.6) satisfies lim J−(t; z, z) = 0 for every z C ([9, Theorem∈ 40]). t→t1 ∈ + From the above, if we assume that RH and SC hold, there exists a Hamiltonian Hξ defined on some interval I such that Eξ of (1.2) is recovered from a solution of the canonical system associated with Hξ. Using the conjectural Hamiltonian Hξ, the above naive strategy to the proof of RH is now refined as follows:
(1-1) Constructing the Hamiltonian Hξ on I = [t0,t1) without RH; t (1-2) Constructing the solution [Aξ(t, z),Bξ(t, z)] of the canonical system on I asso- ciated with H satisfying E (z)= A (t , z) iB (t , z); ξ ξ ξ 0 − ξ 0 (1-3) Showing that limt→t1 Jξ(t; z, z) = 0 for every z C+, where Jξ(t; z, w) is the t ∈ function defined by (1.6) for [Aξ(t, z),Bξ (t, z)].
We can conclude that Eξ belongs to HB if the above three steps are completed, and hence all zeros of ξ(1/2 iz)= A (z) = (E (z)+ E♯(z))/2 are real; this is nothing but − ξ ξ ξ HAMILTONIANS ARISING FROM L-FUNCTIONSIN THESELBERG CLASS 3
RH. However, this approach faces a serious obstacle from the first step, because the inverse problem of canonical systems, which explicitly recovers H from a given E HB, is difficult in general. As mentioned in the introduction to [45], there are several methods∈ for constructing a Hamiltonian H for a large class of E HB, but they are inconvenient to use in the above strategy, because such methods are usuall∈ y not applicable if it is not known whether E HB. In order to avoid∈ such obstacles, we consider the family of entire functions Eω(z) := ξ( 1 + ω iz), ω> 0 ξ 2 − instead of the single function E . (Note that E = E0, but Eω for small ω > 0 is similar ξ ξ 6 ξ ξ to E in the sense that Eω(z) = ξ( 1 + ω iz)+ ωξ′( 1 + ω iz)+ O(ω2) for small ξ ξ 2 − 2 − ω > 0 if z in a compact set, where “O” is the Landau symbol.) Then we find that a ω necessary and sufficient condition for RH (not require SC) is that Eξ HB for every ∈ ω ω > 0 (Propositions 2.1 and 2.2). In particular, there exists a Hamiltonian Hξ for every ω > 0 under RH. Therefore, RH is proved by completing the following three steps for every ω > 0: ω (2-1) Constructing the Hamiltonian Hξ on I = [t0,t1) without RH; t ω ω (2-2) Constructing the solution [Aξ (t, z),Bξ (t, z)] of the canonical system on I asso- ciated with Hω satisfying Eω(z)= Aω(t , z) iBω(t , z); ξ ξ ξ 0 − ξ 0 (2-3) Showing that lim J ω(t; z, z) = 0 for all z C . t→t1 ξ ∈ + There are two advantages to the second strategy. The first advantage is that a Hamil- tonian Hω on I = [0, ) and a solution t[Aω(t, z),Bω(t, z)] of the associate canonical ξ ∞ ξ ξ system are explicitly constructed in [43] under the restriction ω > 1 by applying the method of Burnol [13] introduced for the study of the Hankel transform. The second ad- vantage is the avoiding of SC and the central zero, that is, multiple zeros on the critical line and the zero at the central point s = 1/2 are allowed to ξ in the second strat- egy. This point is important to generalize the above strategy to the other L-functions, because they often have a multiple zero at the central point s = 1/2. ω t ω ω ± In [43], Hξ and [Aξ (t, z),Bξ (t, z)] are constructed by using solutions ϕt of the integral equations ϕ±(x) t Kω(x + y)ϕ±(y) dy = Kω(x + t), where Kω is the kernel t ± −∞ ξ t ξ ξ defined by R 1 (Eω)♯(z) Kω(x)= ξ e−izx dz ξ 2π Eω(z) Zℑ(z)=c ξ for large c > 0. The behavior of t[Aω(t, z),Bω(t, z)] at t = and its role in the ξ ξ ∞ proof of Eω HB were not studied in [43]. However, if the construction of Hω and ξ ∈ ξ t[Aω(t, z),Bω(t, z)] is extended to 0 < ω 1 together with an additional reasonable ξ ξ ≤ result on the behavior of t[Aω(t, z),Bω(t, z)] at t = , we obtain Eω HB for every ξ ξ ∞ ξ ∈ ω > 0, which implies RH. Unfortunately, there were several technical difficulties in [43] ω t ω ω to extend the construction of Hξ and [Aξ (t, z),Bξ (t, z)] to 0 < ω 1. For instance, ω 2 ≤ KL is far from continuous functions and L -functions if ω > 0 is small, and this fact is a serious obstacle for the construction in [43]. In this paper, we resolve the above technical difficulties by introducing the additional discrete parameter ν: Eω,ν (z)= ξ( 1 + ω iz)ν , ω R , ν Z . ξ 2 − ∈ >0 ∈ >0 ω,ν The parameter ν does not affect to study whether EL HB by definition of HB, but ∈ ω,ν plays an important role in constructing the conjectural Hamiltonian Hξ obtained by ω,ν ω,ν applying the method in [45] to Eξ . We will show in Section 4 that EL satisfies all conditions required for the method in [45] due to the existence of parameter ν. In this 4 M. SUZUKI way, a large part of (2-1) and (2-2) are achieved successfully for each ω > 0. To study (2-3), we need more preparation, as described in Section 2.
Summarizing the above discussion, we will obtain an equivalent condition for RH in terms of canonical systems associated with Hamiltonians. This framework to establish the equivalent condition for RH applies to more general zeta and L-functions. We apply it to L-functions in the Selberg class which was introduced in Selberg [39] together with a sophisticated consideration about the question of what is an L-function. Then we obtain an equivalent condition for an analogue of RH for L-functions in the Selberg class as in Section 2. This is the goal of this paper.
Before concluding the introduction, we comment on the Hilbert-P´olya conjecture, a conjectural possible approach to RH. It claims that the imaginary parts of the nontrivial zeros of ζ(s) are eigenvalues of some unbounded self-adjoint operator D acting on a Hilbert space . The Montgomery-Odlyzko conjecture on the vertical distribution of the nontrivial zerosH of ζ(s) and the resemblance between the Weil explicit formula and the Selberg trace formula are strong evidence to the Hilbert-P´olya conjecture. No pair ( , D) of space and operator had been found, although the conjectural pair were suggestedH by several authors. Among them, the idea of Connes [14] for the conjectural Hilbert-P´olya pair ( , D) is very attractive in the sense that it stands on the local–global principle in numberH theory (adeles and ideles), it enables us to understand the Weil explicit formula as a trace formula, and it is stated not only for ζ(s) but also Dedekind zeta-functions and Hecke L-functions. In addition, his idea is compatible with the Berry–Keating model [2] which is an attempt to explain RH by using a physical model. If Eω,ν HB, we can construct a family of pairs ( ω,ν(ω), Dω,ν(ω)) of Hilbert ξ ∈ { H }ω>0 spaces and self-adjoint operators. The family may be regarded as a possible realization of Connes’ Hilbert-P´olya pair ( , D) by allowing the perturbation parameter ω. This topic will be treated more preciselyH in Section 7.
The paper is organized as follows. In Section 2, we state the main results Theorem 2.1, Theorem 2.2, Theorem 2.3 and Theorem 2.4 after a small preparation of notation. The first two theorems associate the above inverse problem of canonical systems to GRH. The third theorem is related to the necessity of GRH in the equivalent condition in the fourth theorem. The fourth theorem is the goal of this paper which is an equivalent condition for GRH in terms of the inverse problem of canonical systems. In Section 3, we briefly review the method of [45] for the inverse problem of canonical systems to use it in the proof of the main results. In Section 4, we prove that [45, Theorem 1.1] applies ω,ν ω,ν to Eξ and its generalization EL to L-functions in the Selberg class. The proof of Theorem 2.1 is a part of this process. Then, we obtain Theorem 2.2 by applying [45, ω,ν Theorem 1.1] to EL . In Section 5, we prove Theorem 2.3 by using the theory of de Branges spaces. In Section 6, we prove Theorem 2.4 by combining several results in former sections. In Section 7, we comment on the Hilbert-P´olya conjecture and Connes’ approach about it from the viewpoint of the theory of canonical systems. In Section 8, we give miscellaneous remarks on the results and contents of this paper. Basically, we attempt as much as possible to prove the main results in Section 2 by applying general results to L-functions in the Selberg class for the convenience of applications to other class of L-functions.
Acknowledgments This work was supported by JSPS KAKENHI Grant Number JP25800007 and JP17K05163. Also, this work was partially supported by French- Japanese Projects “Zeta Functions of Several Variables and Applications” in Japan- France Research Cooperative Program supported by JSPS and CNRS. HAMILTONIANS ARISING FROM L-FUNCTIONSIN THESELBERG CLASS 5
2. Main Results 2.1. Selberg class and GRH. Let s = σ + it be the complex variable. The Selberg class consists of the Dirichlet series S ∞ a (n) L(s)= L (2.1) ns nX=1 satisfying the following five axioms2: (S1) The Dirichlet series (2.1) converges absolutely if σ > 1. (S2) Analytic continuation – There exists an integer m 0 such that (s 1)mL(s) extends to an entire function of finite order. ≥ − (S3) Functional equation – L satisfies the functional equation Λ (s)= ǫ Λ♯ (1 s), L L L − where r Λ (s)= Qs Γ(λ s + µ ) L(s)= γ (s) L(s), L j j · L · jY=1 Γ is the gamma function and r 0, Q > 0, λ > 0, µ C with (µ ) 0, ≥ j j ∈ ℜ j ≥ ǫL C with ǫL = 1 are parameters depending on L. ∈ | | ε (S4) Ramanujan conjecture – For every ε> 0, aL(n) ε n . (S5) Euler product – For every sufficiently large σ, ≪ ∞ b (n) log L(s)= L , ns n=1 X where b (n) = 0 unless n = pm with m 1, and b (n) nθ for some θ < 1/2. L ≥ L ≪ The Riemann zeta-function and Dirichlet L-functions associated with primitive Dirichlet characters are typical members of the Selberg class . As with these examples, it is conjectured that all major zeta- and L-functions appearingS in number theory, such as automorphic L-functions and Artin L-functions, are members of . Considering this conjecture, is a proper class of L-functions in studying the analogueS of RH for number theoretic L-functions.S See the survey article [35] of A. Perelli for an overview of results, conjectures, and problems relating to the Selberg class. We define the degree d of L by d = 2 r λ , where λ are numbers in (S3). L ∈ S L j=1 j j From (S5), we have a (1) = 1 and find that coefficients a (n) define a multiplicative L P L arithmetic function. From (S3) and (S5), L has no zeros outside the critical strip 0 σ 1 except for zeros in the half-plane∈ Sσ 0 located at poles of the involved gamma≤ ≤ factors. The zeros lie in the critical strip≤ are called the nontrivial zeros. The nontrivial zeros are infinitely many unless L 1 and coincide with the zeros of the entire function3 ≡ ξ (s)= smL (s 1)mL Λ (s) (2.2) L − L of order one, where mL is the minimal nonnegative integer m in (S2). It is conjectured that the analogue of RH holds for all L-functions in : S Grand Riemann Hypothesis (GRH). For L , ξ (s) = 0 unless (s) = 1/2. ∈ S L 6 ℜ For L , we abbreviate to GRH(L) the assertion that all zeros of ξL(s) lie on the critical∈ line S (s) = 1/2. ℜ 2 As a matter of fact, (S1) is unnecessary to define S because it is derived from (S4). But we put it into the axiom of S according to other literature. A positive reason is that it is convenient to define the extended Selberg class which is the class of Dirichlet series satisfying (S1)∼(S3). 3 r − 1 The quantity 2 Pj=1(µj 2 ) is usually referred to as ξ-invariant and is often written as ξL in the theory of the Selberg class, but we do not use the letter ξ for the ξ-invariant of L ∈S to avoid confusion. 6 M. SUZUKI
The main subject of this paper is the study of GRH(L) for L-functions in the subclass R of defined by S S R = L L 1, L(R) R, Λ (R) R . S { ∈ S | 6≡ ⊂ L ⊂ } If L R, the gamma factor γL in (S3) is not an exponential function, ξL are non- constant∈ S entire functions satisfying functional equations ξ (s)= ǫ ξ (1 s) and ξ (s)= ξ♯ (s), (2.3) L L L − L L where the root number ǫL must be 1 or 1. The second equality of (2.3) means that ξL is a real entire function, which means an− entire function F satisfying F = F ♯. 2.2. Auxiliary functions for GRH. Let z = u + iv be the complex variables relating to the variable s by s = 1/2 iz. To work on GRH(L) for L R, we introduce the two parameter family of entire− functions ∈ S Eω,ν (z) := ξ ( 1 + ω iz)ν = ξ (s + ω)ν (2.4) L L 2 − L parametrized by ω R and ν Z . Then, ∈ >0 ∈ >0 (Eω,ν )♯(z)= Eω,ν( z) L L − ω,ν ♯ ν 1 ν by the second equation in (2.3), and then (EL ) (z) = ǫL ξL( 2 ω iz) by the first ω,ν ω,ν− − equation in (2.3). Therefore, real entire functions AL and BL defined by (1.4) for ω,ν E = EL are even and odd, respectively. The following proposition is trivial from definition (2.4) and functional equations (2.3). ω,1 Proposition 2.1. For L R, GRH(L) holds if E HB for all ω > 0. ∈ S L ∈ ω,ν As easily found from (2.4) and definition of HB, for fixed ω > 0, if EL HB for some ω,ν ω,1 ∈ ν Z>0, then EL HB for arbitrary ν Z>0. Therefore, EL in Proposition 2.1 can ∈ ω,ν∈ω ∈ be replaced by EL defined for positive integers νω indexed by ω > 0. On the other hand, HB in the statement can be replaced by HB without the changing of the meaning ω,1 ω,1 of the statement, although EL HB is different from EL HB for individual ω > 0. As the converse of Proposition∈ 2.1, we have the following.∈ ω,ν Proposition 2.2. Let L R and ν Z>0. Then, EL belongs to HB for every ω > 1/2 unconditionally and∈ forS every 0 ∈<ω 1/2 under GRH(L). ≤ ω,ν Proof. First, we suppose that ω > 1/2. Then EL has no real zeros, since all zeros of ω,ν ξL(s) lie in the vertical strip 0 σ 1. On the other hand, we find that EL satisfies ≤ ≤ ν (1.3) by applying [30, Theorem 4] to ξL(s) , since ξL(s) satisfies (2.3) and has only zeros ω,ν in the strip 0 σ 1. Hence EL HB. The case of 0 <ω 1/2 is proved similarly under GRH(L≤). ≤ ∈ ≤ The value ω = 1/2 in Proposition 2.2 comes from the trivial zero-free region of L . ω,ν ∈ S More precise relation between the zero-free region of L and the property EL HB will be discussed in Proposition 4.3. ∈
From Propositions 2.1 and 2.2, the validity of GRH(L) for L R is equivalent to ∈ Sω,ν the existence of a section R>0 R>0 Z>0; ω (ω,ν) such that EL HB for every ω > 0. Hence GRH(L) will be→ established× if the7→ following three steps are∈ completed for every point of a section: ω,ν ω,ν (3-1) Constructing the Hamiltonian HL on I = [t0,t1) from EL without GRH(L); t ω,ν ω,ν (3-2) Constructing the solution [AL (t, z),BL (t, z)] of the canonical system on I ω,ν ω,ν ω,ν ω,ν associated with HL satisfying EL (z)= AL (t0, z) iBL (t0, z); ω,ν − ω,ν (3-3) Showing that limt→t1 JL (t; z, z) = 0 for all z C+, where JL (t; z, w) is the t ω,ν ω,ν ∈ function defined by (1.6) for [AL (t, z),BL (t, z)]. The main results stated below concern each step in (3-1) (3-3). ∼ HAMILTONIANS ARISING FROM L-FUNCTIONSIN THESELBERG CLASS 7
ω,ν 2.3. Results on (3-1). We define the function KL on the real line by
ω,ν ♯ ω,ν 1 ν (E ) (z) E ( z) ξL( ω iz) Θω,ν(z) := L = L − = ǫν 2 − − (2.5) L Eω,ν(z) Eω,ν(z) L ξ ( 1 + ω iz)ν L L L 2 − and 1 ∞ Kω,ν(x) := Θω,ν(u + iv) e−ix(u+iv) du (2.6) L 2π L Z−∞ for L R and (ω,ν) R>0 Z>0. The integral on the right-hand side of (2.6) converges absolutely∈ S if v > 0 is∈ sufficiently× large, and we have
⌊exp(x)⌋ qω,ν(n) Kω,ν(x)= ǫν L Gω,ν(x log n) (2.7) L L √n L − nX=1 ω,ν ω,ν for x > 0 and KL (x) = 0 for x < 0, where ǫL is the root number in (2.3), qL (n) is an arithmetic function determined by the Dirichlet coefficients aL(n) (non-archimedean information), and Gω,ν is a certain explicit real-valued function having support in [0, ) L ∞ determined by the gamma factor γL (archimedean information). We find that Kω,ν is a continuous function if (ω,ν) R Z satisfies L ∈ >0 × >0 νωdL > 1, (2.8) where dL is the degree of L R (Proposition 4.1). This condition for (ω,ν) is technical but essential to the following∈ constructionS of Hω,ν. If(ω,ν) R Z satisfies (2.8), L ∈ >0 × >0 t Kω,ν[t] : f(x) 1 (x) Kω,ν(x + y) f(y) dy (2.9) L 7→ (−∞,t) L Z−∞ 2 defines a bounded operator on L ( ,t) for every t R, where 1A is the characteristic K−∞ω,ν ∈ function of a set A. The study of L [t] yields a canonical system as follows.
Theorem 2.1. Let L R and (ω,ν) R>0 Z>0 be such that (2.8) holds. Then, for every t> 0, (2.9) defines∈ S a Hilbert-Schmidt∈ × type self-adjoint operator on L2( ,t) Kω,ν −∞ having a continuous kernel, and L [t] = 0 for t 0. Moreover, there exists τ = ≤ Kω,ν τ(L; ω,ν) > 0 such that both 1 are not the eigenvalues of L [t] for every t [0,τ). In particular, the Fredholm determinant± det(1 Kω,ν [t]) does not vanish for every∈t [0,τ). ± L ∈ ω,ν The first half will be proved by applying the argument of [45, Section 1] to KL under Proposition 4.1. The latter half is Proposition 4.2. By (2.9), to understand the Kω,ν ω,ν operator L [t], we need only the values of KL on [0, 2t) that are determined by the ω,ν information on the gamma factor γL and finitely many coefficient qL (n)’s by (2.7). ω,ν By Theorem 2.1, the Hamiltonian HL on [0,τ) is defined by ω,ν Kω,ν 2 ω,ν 1/γL (t) 0 ω,ν det(1 + L [t]) H (t) := ω,ν , γ (t) := , (2.10) L 0 γ (t) L det(1 Kω,ν [t]) L − L and it defines the canonical system
d A(t, z) 0 1 ω,ν A(t, z) = z − H (t) , z C (2.11) − dt B(t, z) 1 0 L B(t, z) ∈ on [0,τ). Next, we construct a unique solution of the canonical system recovering the ω,ν entire function EL . 8 M. SUZUKI
2.4. Results on (3-2). Let τ be the number in Theorem 2.1. Then, two integral equations (1 Kω,ν[t])ϕ±(x)= 1 (x)Kω,ν(x + t) ± L t (−∞,t) L ± 2 for unknown functions ϕt L ( ,t) have unique solutions for every t [0,τ), since ω,ν ∈ −∞ ∈ 1 K [t] are invertible. We extend the solutions ϕ± to continuous functions on R by ± L t t φ±(t,x)= Kω,ν(x + t) Kω,ν(x + y)ϕ±(y) dy. L ∓ L t Z−∞ Then, we find that φ±(t,x) ec|x| for some c> 0 which may depend on t (Proposition 4.1 and [45, Lemma| 2.7]). Therefore,|≪ the two functions defined by 1 ∞ Aω,ν(t, z) := γω,ν(t)1/2Eω,ν (z) eizt + φ+(t,x)eizx dx , L 2 L L Zt (2.12) 1 ∞ iBω,ν(t, z) := γω,ν(t)−1/2Eω,ν (z) eizt φ−(t,x)eizx dx − L 2 L L − Zt are analytic in the upper half-plane (z) > c′ for t [0,τ). ℑ ∈ Theorem 2.2. Let L R and let (ω,ν) R>0 Z>0 be such that (2.8) holds. We define ω,ν ∈ S ω,ν ∈ ω,ν × ω,ν EL by (2.4), and then define AL and BL by (1.4) for E = EL . Let τ = τ(L; ω,ν) ω,ν ω,ν be the positive real number in Theorem 2.1. Let AL (t, z) and BL (t, z) be the families of functions defined by (2.12). Then, ω,ν ω,ν (1) AL (t, z) and BL (t, z) extend to real entire functions as a function of z for every t [0,τ), ω,ν ∈ ω,ν (2) AL (t, z) is even and BL (t, z) is odd as a function of z for every t [0,τ), ω,ν ω,ν ∈ (3) AL (t, z) and BL (t, z) are continuous and piecewise continuously differentiable functions as functions of t for every z C, t ω,ν ω,ν ∈ (4) [AL (t, z),BL (t, z)] solves the canonical system (2.11) on [0,τ) associated with ω,ν the Hamiltonian HL , ω,ν ω,ν ω,ν ω,ν (5) AL (0, z)= AL (z), BL (0, z)= BL (z), and thus Eω,ν(z)= Aω,ν(0, z) iBω,ν(0, z). (2.13) L L − L Theorem 2.2 is a generalization of [43, Theorem 2.3] which only deal with the case of L = ζ and ν = 1. The condition (2.8) is ω > 1 for L = ζ and ν =1 by dζ = 1. ω,ν Theorem 2.2 will be proved by applying [45, Theorem 1.1] to the entire function EL under Propositions 4.1 and 4.2 that are proved in Section 4.
ω,ν 2.5. Results on (3-3). By the above results, we obtain the Hamiltonian HL defined t ω,ν ω,ν on [0,τ) and the solution [AL (t, z),BL (t, z)] of the canonical system associated with ω,ν HL satisfying (2.13) without GRH(L) for L R. On the other hand, [9, Theorem 40] and [43, Theorem 2.3] suggest that their domain∈ S extends to [0, ) under GRH(L). ω,ν ∞ Recall that GRH(L) implies EL HB for every (ω,ν) by Proposition 2.2. If we ω,ν∈ ω,ν suppose the weaker condition EL HB, the Hamiltonian HL of (2.10) and the t ω,ν ω,ν ∈ solution [AL (t, z),BL (t, z)] of (2.12) extend to I = [0, ), because the Fredholm Kω,ν ∞ determinant det(1 L [t]) does not vanish for every t 0 (Proposition 5.1). The following result asserts± that the condition in (3-3) with t =≥ is necessary for GRH(L). 1 ∞ Theorem 2.3. Let L R and let (ω,ν) R>0 Z>0 be such that (2.8) holds. Assume ω,ν ∈ S ∈ ω,ν× ω,ν that EL HB if 0 < ω 1/2 and define JL (t; z, w) by (1.6) using AL (t, z) and ω,ν ∈ ≤ ω,ν BL (t, z) of Theorem 2.1 for t 0. Then JL (t; z, z) 0 as a function of z for any t 0 and lim J ω,ν(t; z, w) = 0 for≥ every fixed z, w C. 6≡ ≥ t→∞ L ∈ HAMILTONIANS ARISING FROM L-FUNCTIONSIN THESELBERG CLASS 9
2.6. Equivalent condition for GRH. The condition in (3-3) with t1 = is a suffi- ω,ν ω,ν ∞ cient condition for EL HB by [45, Proposition 2.4]. Therefore, EL HB is equiv- ω,ν ∈ ω,ν alent that H is extended to a Hamiltonian on [0, ) and lim J ∈(t; z, w)=0if L ∞ t→∞ L νωdL > 1. Hence we obtain the following equivalent condition for GRH(L) for L R by noting that the range of ω in Proposition 2.1 can be relaxed to a decreasing sequence∈ S tending to zero.
Theorem 2.4. The validity of GRH(L) for L R is equivalent to the condition that there exists a sequence (ω ,ν ) R Z , n∈ S1, such that n n ∈ >0 × >0 ≥ (1) ω <ω if m>n and ω 0 as n , m n n → →∞ (2) νnωndL > 1, Kωn,νn (3) det(1 L [t]) = 0 for every t 0, and ±ωn,νn 6 ≥ (4) lim J (t; z, z) = 0 for every z C+. t→∞ L ∈ The detailed proof of Theorem 2.4 will be given in Section 6. Also, a variant of Theorem 2.4 is stated in Section 6.3.
3. Inverse problem for some special canonical system We review the way of construction of Hamiltonians in [45]. As usual, we denote by
∞ 1 ∞ (Ff)(z)= f(x) eixz dx, (F−1g)(z)= g(u) e−ixu du 2π Z−∞ Z−∞ the Fourier integral and inverse Fourier integral, respectively, and use the Vinogradov symbol “ ” to estimate a function as well as the Landau symbol “O”. Let E ≪be an entire function satisfying the following five conditions: (K1) there exists a real-valued continuously differentiable function ̺ on R such that ̺(x) e−n|x| for any n> 0 and E(z) = (F̺)(z) for all z C; (K2) there| |≪ exists a real-valued continuous function K on R ∈such that K(x) exp(c x ) for some c 0 and E♯(z)/E(z) = (FK)(z) holds for (z) >| c; | ≪ (K3) K vanishes| | on ( ,≥0); ℑ (K4) K is continuously−∞ differentiable outside a discrete subset Λ R and K′ is locally integrable on R; ⊂ | | (K5) there exists 0 < τ such that both 1 are not eigenvalues of the operator ≤ ∞ ± K[t] : f(x) 1 (x) t K(x + y) f(y) dy on L2( ,t) for every 0 t<τ. 7→ (−∞,t) −∞ −∞ ≤ Then, the matrix-valued functionR H on [0,τ) defined by
1/γ(t) 0 det(1 + K[t]) 2 H(t) := , γ(t)= 0 γ(t) det(1 K[t]) − is a Hamiltonian on [0,τ). We can also construct a unique solution t[A(t, z),B(t, z)] of the canonical system associated with this H satisfying (A(0, z),B(0, z)) = (A(z),B(z)) as in [45, Theorem 1.1] by a way similar to [13] and [43], where A and B are entire functions defined by (1.4).
4. Proof of Theorems 2.1 and 2.2 ω,ν In this section, we prove that EL of (2.4) satisfies (K1) (K5) in Section 3 (Propo- sitions 4.1 and 4.2). Then, we obtain Theorem 2.1 as a consequ∼ ence of Propositions 4.1 and 4.2. Also, we obtain Theorem 2.2 as a consequence of [45, Theorem 1.1]. 10 M. SUZUKI
ω,ν ω,ν 4.1. Analytic properties of EL and ΘL . ω,ν Lemma 4.1. The entire function EL define by (2.4) for L R and (ω,ν) R>0 Z>0 satisfies (K1). More precisely, if we define ∈ S ∈ ×
1 ̺ω,ν(x)= Eω,ν(z)e−izxdz (4.1) L 2π L Zℑ(z)=c by taking some c R, then it is a real-valued function in C∞(R) satisfying the estimate ω,ν ∈ ̺ (x) e−n|x| for every n N such that | L |≪n ∈ ∞ ω,ν F ω,ν ω,ν izx EL (z) = ( ̺L )(z)= ̺L (x)e dx Z−∞ holds for all z C. ∈ Proof. From (S1), L(s) is bounded in (s) 2. From (S2), (S3) and the Phragm´en– ℜ ≥ Lindel¨of principle, (s 1)mL L(s) is bounded by a polynomial of s in any vertical strip − −n of finite width. Thus, ξL(s) decays faster than s for any n N in any vertical strip ω,ν | | ∈ ∞ of finite width. Hence ̺L is defined by (4.1) independent of c and belongs to C (R). ω,ν ♯ ω,ν From the second equation of (2.3), we have (EL ) (z) = EL ( z). Thus, by taking ω,ν − c = 0 in (4.1), we find that ̺L is real-valued. By moving the path of integration in ω,ν −n|x| (4.1) to the upside or downside, we see that ̺L (x) n e for every n N. Taking ω,ν| |≪F ω,ν ∈ the Fourier transform of (4.1), we obtain EL (z) = ( ̺L )(z). ω,ν Lemma 4.2. Let ΘL be the meromorphic function in C define by (2.5) for L R and (ω,ν) R Z . Then the estimate ∈ S ∈ >0 × >0 Θω,ν(u + iv) (1 + u )−νωdL (4.2) | L |≪v | | holds for any fixed v> 1/2+ ω, Moreover, the estimate
Θω,ν(u + iv) (1 + v)−νωdL (4.3) | L |≪δ holds uniformly for u R and v 1/2+ ω + δ for any fixed δ > 0. ∈ ≥ ∞ Proof. We have ξL(s)= ξL (s)L(s) by putting
r ξ∞(s)= smL (s 1)mL γ (s)= smL (s 1)mL Qs Γ(λ s + µ ). (4.4) L − · L − · j j jY=1 Then, in order to prove (4.2) and (4.3), it is sufficient to prove
ξ∞(s ω) ξ∞(s ω) L − (1 + u )−ωdL and L − (1 + v)−ωdL (4.5) ξ∞(s + ω) ≪ | | ξ∞(s + ω) ≪ L L
for s = 1/2 i(u+iv) with v 1/2+ω +δ by (2.2) and (2.5), since L(s ω)/L(s+ω) is expressed as− an absolutely convergent≥ Dirichlet series in a right-half plane− (s) > 1+ ω and hence it is bounded there. ℜ Let λ> 0, µ = µ + iµ C with µ 0. Using the Stirling formula 0 1 ∈ 0 ≥ 2π s s Γ(s)= (1 + O ( s −1)) s 1, arg s <π δ s e δ | | | |≥ | | − r HAMILTONIANS ARISING FROM L-FUNCTIONS IN THE SELBERG CLASS 11 for δ = π/4 and s = 1/2 i(u + iv) (u R, v > 0), we have − ∈ Γ(λ(s ω)+ µ) 2λω 1+ O( λ(s ω)+ µ −1) − = 1+ | − | e2λω Γ(λ(s + ω)+ µ) λ(s ω)+ µ · 1+ O( λ(s + ω)+ µ −1) · s − | | exp 2λω log λ( 1 + v ω)+ µ i(λu µ ) × − 2 − 0 − − 1 h 1 2λω + λ( 2 + v + ω)+ µ0 log 1 1 − λ( + v + ω)+ µ0 i(λu µ1) 2 − − 2λω + (λu µ1) arg 1 1 . − − λ( + v + ω)+ µ0 i(λu µ1)! 2 − − i On the right-hand side,
2λω 1+ O( λ(s ω)+ µ −1) 1+ | − | e2λω, λ(s ω)+ µ · 1+ O( λ(s + ω)+ µ −1) · s − | |
1 2λω λ( 2 + v + ω)+ µ0 log 1 1 − λ( + v + ω)+ µ0 i(λu µ1) 2 − − and 2λω (λu µ1) arg 1 − − λ( 1 + v + ω)+ µ i(λu µ ) 2 0 − − 1 ! are uniformly bounded for u R and v 1/2+ ω + δ. Hence ∈ ≥ Γ(λ(s ω)+ µ) − exp 2λω log λ( 1 + v ω)+ µ i(λu µ ) , (4.6) Γ(λ(s + ω)+ µ) ≪ − 2 − 1 − − 2
h i where the implied constant depends on ω, λ , µ and δ > 0. This implies the first estimate of (4.5) by (4.4) and the definition of dL. On the other hand, the right-hand side of (4.6) takes the maximum [λ( 1 + v ω)+ µ ]−2λω at u = µ /λ as a function of u. Hence 2 − 1 1 Γ(λ(s ω)+ µ) − (1 + v)−2λω Γ(λ(s + ω)+ µ) ≪
holds uniformly for u R and v 1/2+ ω + δ, where the implied constant depends on ω, λ, µ and δ > 0. This∈ implies the≥ second estimate of (4.5) by (4.4) and the definition of dL.
Let a be the arithmetic function defined by the Dirichlet series (2.1) of L R. L ∈ S Then, by a (1) = 1 = 0, the Dirichlet inverse a−1 exists and is given by a−1(n) = L 6 L L a (n/d)a−1(d) for n> 1 and a−1(1) = 1. We have − d|n, d 1 1 1 ω− − µ − 2λω−1 g (y)= y 2 λ (1 y λ ) ω,λ,µ λΓ(2λω) − 12 M. SUZUKI for y > 1 and gω,λ,µ(y)=0 for 0 Proof. Properties (1) and (2) are trivial by definition, and (5) is a simple consequence of (2.6) and (4.2). To prove (3), we use the variable s = 1/2 iz for convenience. If ω > 0, λ> 0, (µ) 0, we have − ℜ ≥ ∞ 1 1 1 Γ(λ(s ω)+ µ) 1 ω− − µ − 2λω−1 −s dy − = y 2 λ (1 y λ ) y 2 (4.10) Γ(λ(s + ω)+ µ) λΓ(2λω) − y Z1 for (s) >ω (µ)/λ by [34, (5.35) of p.195]. Therefore, we obtain ℜ −ℜ ν ∞ r 1 dy Γ(λ (s ω)+ µ ) ω,ν 2 −s −2ω j j g˜L (y) y = Q − (4.11) 0 · y Γ(λj(s + ω)+ µj) Z jY=1 for (s) > max1≤j≤r(ω (µj)/λj ) by applying [47, Theorem 44] repeatedly to (4.10). Applyingℜ the formula−ℜ 1 ∞ 1 1 dy 1 y−a− 2 (log y)k−1 y 2 −s = , (s + a) > 0 (k 1)! · y (s + a)k ℜ Z1 ω,ν − to rL , we have ∞ νmL ω,ν 1 dy (s ω)(s ω 1) r (y) y 2 −s = − − − (4.12) L · y (s + ω)(s + ω 1) Z1 − for (s) > 1 ω (we assumed ω > 0). Thenℜ we obtain− ∞ 1 dy ∞ 1 dy ω,ν 2 −s ω,ν 2 −s gL (y) y = gL (y) y 0 · y 1 · y Z Z ν νmL r ∞ ν (s ω)(s ω 1) −2ω Γ(λj(s ω)+ µj) ξL (s ω) = − − − Q − = ∞ − (s + ω)(s + ω 1) Γ(λj(s + ω)+ µj) ξL (s + ω) − jY=1 for (s) > max(1 ω, max1≤j≤r(ω (µj)/λj )) by applying [47, Theorem 44] to (4.11) andℜ (4.12). On the− other hand, we−ℜ have ∞ ∞ L(s ω) ν aν (m)mω a−ν(n)n−ω − = L L L(s + ω) ms ns m=1 n=1 X X ∞ ∞ 1 aν (n/d)a−ν (d) qω,ν(n) = nω L L = L ns d2ω ns n=1 n=1 X Xd|n X by definition (4.7), where the series converges absolutely for(s) > 1+ ω. By definition (4.8), we have formally ℜ ∞ ω,ν ∞ 1 dy c (n) ∞ 1 dy kω,ν(y) y 2 −s = ǫν L gω,ν (y/n) (y/n) 2 −s L y L ns L y 0 n=1 0 Z X Z ξ∞(s ω) L(s ω) ν = ǫν L − − , L ξ∞(s + ω) L(s + ω) L and it is justified by Fubini’s theorem for (s) > 1+ ω. By the changing of variables y = ex and s = 1/2 iz, we obtain (4.9) andℜ complete the proof of (3). − ω,ν We prove (4) and the last line of Proposition 4.1. By (4.2), ΘL (u + iv) belongs to 1 ω,ν L (R) as a function of u if v is sufficiently large. Thus, KL is uniformly continuous on R by (2.6). Moreover, by (2.6) and (4.2), the formula dn 1 Kω,ν(x)= Θω,ν(z)( iz)n e−izx dz dxn L 2π L − Zℑ(z)=c holds together with the absolute convergence of the integral on the right-hand side. ω,ν k Therefore, this shows that KL is C if νωdL > k + 1. 14 M. SUZUKI d ω,ν Finally, we prove (6). The derivative dx KL is locally integrable by (4). On the d ω,ν other hand, the set of possible singularities of dx KL is discrete in R by (2), and d ω,ν dx KL does not change its sign infinitely often around any possible singularity except for x = + by definition of gω,ν . Therefore, the local integrability of d Kω,ν implies ∞ L dx L the local integrability of d Kω,ν . | dx L | ω,ν By Proposition 4.1, we find that EL satisfies (K1) (K4) if (ω,ν) satisfies (2.8). ω,ν ∼ Successively, we show that EL satisfies (K5) for sufficiently small τ > 0 unconditionally. 4.2. Non-vanishing of Fredholm determinants: Unconditional cases. In this Kω,ν section, we understand that L f is a function defined by the integral ∞ Kω,ν ω,ν ( L f)(x)= KL (x + y) f(y) dy (4.13) Z−∞ if the right-hand side converges absolutely and locally uniform for a function f, because we do not assume that Eω,ν HB (which implies that Kω,ν f belongs to L2(R) for every L ∈ L f L2(R) by [45, Lemmas 2.1 and 2.2]). ∈ Proposition 4.2. Let L R. Suppose that (ω,ν) satisfies (2.8) and define the operator Kω,ν 2 ∈ S L [t] on L ( ,t) by (2.9). Then, there exists τ = τ(L; ω,ν) > 0 such that both 1 −∞ Kω,ν Kω,ν ± are not eigenvalues of L [t] for every 0 t<τ, that is, both 1 L [t] are invertible ≤ ω,ν ± operators on L2( ,t) for every 0 t<τ. Thus E satisfies (K5) for [0,τ). −∞ ≤ L Kω,ν Kω,ν Proof. The spectrum of L [t] is discrete and consists of eigenvalues, since L [t] is a Hilbert-Schmidt operator on L2( ,t) by (K2) and (K3). The statement of the Kω,ν −∞ 2 Kω,ν proposition is equivalent that L [t]f = f for any f L ( ,t), because 1 µ L [t] 6 ± ∈Kω,ν −∞ − is invertible if 1/µ is not an eigenvalue. In addition, L [t]f = f is equivalent that ω,ν 6 ± P K f = f, since P f = f for f L2( ,t), where P is the orthogonal projection t L 6 ± t ∈ −∞ t from L2(R) to L2( ,t). Suppose that P −∞Kω,νf = f for some 0 = f L2( ,t). We have t L ± 6 ∈ −∞ t P Kω,ν ω,ν ( t L f)(x)= 1(−∞,t)(x) KL (x + y)f(y) dy = 0 Z−∞ 2 P Kω,ν for Proposition 4.3. Let L R, ω0 0 and ν Z>0. Then the following statements are equivalent: ∈ S ≥ ∈ 1 (1) L(s) = 0 for (s) > 2 + ω0, ω,ν 6 ℜ (2) EL belongs to HB for every ω>ω0, ω,ν (3) ΘL is a meromorphic inner function in C+ for every ω>ω0. The value of ν does not affect the above equivalence. Proof. Assume 0 ω0 1/2, since we have nothing to say for ω > 1/2 by Proposition ≤ ≤ ω,ν 2.2 and [45, Lemma 2.1]. We find that (1) implies that EL satisfies (1.3) for every ω>ω in a way similar to the proof of Proposition 2.2. If Eω,ν(z) = ξω( 1 + ω iz)ν 0 L L 2 − has a real zero for some ω>ω , it implies that L(s) has a zero in (s) > 1 + ω , since 0 ℜ 2 0 ξ ( 1 + ω iz)= ξ ( 1 + ω i(z + i(ω ω ))). L 2 − L 2 0 − − 0 ω,ν Thus EL HB and we obtain (1) (2). The implication (2) (3) is a consequence of [45, Lemma∈ 2.1]. The implication⇒ (3) (1) is proved in a way⇒ similar to the proof of Theorem 2.3 (1) in [42]. ⇒ The value ω = 1/2 corresponds to the abscissa σ = 1 of the absolute convergence of the Dirichlet series (2.1). The non-vanishing of L on the line σ = 1 is an important problem because it relates to an analogue of the∈ prime S number theorem of L for example. Conrey–Ghosh [15] proved the non-vanishing of L on the line∈σ S= 1 subject to the truth of the Selberg orthogonality conjecture. Kaczorowski–Perelli∈ S [21] obtained the non-vanishing of L on the line σ = 1 under a weak form of the Selberg orthogonality conjecture. As mentioned∈ S before, it is conjectured that consists only of automorphic L-functions. The non-vanishing for automorphic L-functionsS on the line σ = 1 had been proved unconditionally in Jacquet–Shalika [20]. 5. Proof of Theorem 2.3 ω,ν ω,ν In this section, we prove that EL of (2.4) satisfies (K5) for τ = if EL HB (Proposition 5.1). Then we obtain Theorem 2.3 as a consequence of [45,∞ Theorem∈ 1.2]. 16 M. SUZUKI 5.1. Non-vanishing of Fredholm determinants: Conditional cases. We suppose that Eω,ν HB throughout this subsection, otherwise it will be mentioned. Then, Θω,ν L ∈ L is inner in C+ by [45, Lemma 2.1]. This assumption is satisfied unconditionally for ω > 1/2, and also for all ω > 0 under GRH(L) by Proposition 2.2 (see also Proposition Kω,ν 2 4.3). We denote by L the isometry on L (R) defined by (4.13) (cf. [45, Lemma 2.2]. Note that it is not obvious whether the integral defines an operator on L2(R) if we do not assume that Eω,ν HB.) If (ω,ν) satisfies (2.8), we have L ∈ Kω,ν P Kω,νP L [t]= t L t for Kω,ν[t] in (2.9) and the orthogonal projection P from L2(R) to L2( ,t). L t −∞ ω ω Lemma 5.1. Let L R and ω > 0. Then there exist entire functions f1 (s) and f2 (s) such that they have no∈ S common zeros, satisfy ω ξL(s ω) f2 (s) − = ω , ξL(s + ω) f1 (s) ω and the number of zeros of f2 (s) in (s) T is approximated by c T log T for large T > 0, where c> 0 is some constant. |ℑ | ≤ Proof. We denote by L the set of all zeros of ξL(s) and by m(ρ) the multiplicity of a zero ρ . ThenZ any zero of ξ (s ω) has the form s = ρ + ω for some ρ ∈ ZL L − ∈ ZL and has the multiplicity m(ρ). On the other hand, if s = ρ + ω for some ρ L and ξ (s + ω) = 0, we have ρ + 2ω . Considering this, we set ∈ Z L ∈ZL ω = ρ ρ + 2ω , ZL { ∈ZL | ∈ZL} and define an entire function by the Weierstrass product: m(ρ+2ω) ω s m(ρ + 2ω) s f0 (s)= 1 exp ω − ρ + ω ρ ω ρ∈ZL − m(ρ)≥Ym(ρ+2ω) s m(ρ) m(ρ) s 1 exp , × ω − ρ + ω ρ ω ρ∈ZL − m(ρ) ω ξL(s + ω) ω ξL(s ω) f1 (s)= ω , f2 (s)= ω − . f0 (s) f0 (s) ω ω Then, by definition, f1 (s) and f2 (s) are entire functions such that they have no common zeros and satisfy ω ξL(s ω) f2 (s) − = ω . ξL(s + ω) f1 (s) ω Therefore, the remaining task is to show that f2 (s) has approximately c T log T many zeros in (s) T for some c> 0. |ℑ |≤ ω ω We denote by NL(T ) (resp. NL (T )) the number of zeros in L (resp. L ) with (s) T counting with multiplicity: Z Z |ℑ |≤ ω NL(T )= m(ρ),NL (T )= m(ρ) ρ∈Z , |ℑ(s)|≤T ρ∈Zω, |ℑ(s)|≤T LX L X and define ω nL(T )= m(ρ + 2ω)+ m(ρ), ρ∈ZL, |ℑ(s)|≤T ρ∈ZL, |ℑ(s)|≤T m(ρ)≥Xm(ρ+2ω) m(ρ) ω ω where the first sum is zero if it is an empty sum. Then, nL(T ) NL (T ) NL(T ) and the number of zeros of f ω(s) in (s) T is N (T ) nω(≤T ). We recall≤ that 2 |ℑ | ≤ L − L NL(T ) (dL/π) T log T , and it is so for the number of zeros of ξL(s ω) with (s) T , ∼ − |ℑ ω|≤ where f(T ) g(T ) means that f(T )/g(T ) 1 as T . Therefore, NL(T ) nL(T ) ∼ ω→ →∞ − ∼ c T log T for some c> 0 unless NL(T ) nL(T ). ω ∼ ω Now we prove that NL(T ) nL(T ) by contradiction. Suppose that NL(T ) nL(T ). ω 6∼ω ω ω ω∼ Then, NL(T ) NL (T ), since nL(T ) NL (T ) NL(T ). We put ΣL = ρ L 1 ρ ω and ∼ ≤ ≤ { ∈Z | − ∈ ZL } ω ML (T )= m(ρ). ω ρ∈ΣL,X|ℑ(ρ)|≤T ω ω Then, ML (T ) NL(T ), because NL(T ) NL (T ) and functional equations (2.3) imply ω ∼ ∼ that L is closed under ρ 1 ρ and ρ ρ¯ except for a relatively small subset counting Z 7→ − 7→ω ω ω with multiplicity. If we take a zero ρ ΣL, then 1 ρ L by the definition of ΣL, and ∈ ω − ∈Z thus 1 ρ+2ω L by the definition of L . Therefore, ρ 2ω = 1 (1 ρ+2ω) L by − ∈Z Z − ω − − ∈Zω the first functional equation of (2.3). As a consequence, ρ ΣL implies ρ 2ω L . On ω ω ∈ ω − ∈Z ω the other hand, ML (T ) NL (T ) NL(T ) shows that ρ 2ω L implies ρ 2ω ΣL almost surely. Taken together,∼ ρ∼ Σω implies ρ 2ω− Σ∈Zω almost surely− and∈ this ∈ L − ∈ L process is continued repeatedly. However, it is impossible, because all zeros of ξL(s) must lie in the critical strip. Hence, N (T ) nω(T ). L 6∼ L Kω,νP Lemma 5.2. Let t 0. Suppose that (ω,ν) satisfies (2.8). Then the support of L tf ≥ ω,ν is not compact for every f L2(R) unless K P f = 0. ∈ L t Kω,νP Proof. We prove this by contradiction. Suppose that L tf = 0 and has a compact FKω,νP 6 support. Then L tf is an entire function of exponential type by the Paley-Wiener theorem. On the other hand, we have FKω,νP f(z)=Θω,ν(z) FP f( z) L t L · t − ν by [45, Lemma 2.2]. If we put G(z)= FP f( z)/f ω 1 iz , then we have t − 1 2 − ν FKω,νP f(z)= f ω 1 iz G(z), L t 2 2 − · ω ω where f1 and f2 are functions in Lemma 5.1. HereG(z) is entire, because, by Lemma ω 1 ν ω,ν 5.1, the zeros of the numerator f2 2 iz of ΘL (z) can not kill the zeros of the ω 1 ν − FP denominator f1 2 iz , which therefore must be killed by zeros of af( z). This ν − − allows f ω 1 iz to be factored out. 2 2 The entire− function on the right-hand side has at least c T log T many zeros in the disk of radius T around the origin as T for some c > 0 by Lemma 5.1. However all entire functions of exponential type have→ ∞ at most O(T ) zeros in the disk of radius T around the origin, as T , because of the Jensen formula ([29, 2.5 (15)]). This is a contradiction. →∞ § As the above, it is not necessary to assume that Θ is inner in C+ for Lemma 5.2. Kω,ν 2 Proposition 5.1. Let t 0. We have i) L [t]f = 0 for every f L ( , t), ii) Kω,ν ≥ 2 Kω,ν ∈ −∞ − L [t]f = f for every 0 = f L ( ,t), and iii) L [t] < 1. In particular, k ω,ν k 6 k k 6 ∈ −∞ k k 1 K [t] are invertible operators on L2( ,t) for every t 0. ± L −∞ ≥ Kω,ν 2 Proof. First, we note that L f is defined for every f L ( ,t) by (4.13) and [45, t ω,ν ∈ −∞ Lemma 2.2]). Because −∞ KL (x + y)f(y) dy = 0 for x< t by (K3), we obtain i). Kω,ν − Kω,ν To prove ii), it is sufficient to show L [t]f < f unless f = 0, because L [t] P Kω,ν P R k k k Kkω,ν Kω,ν k k≤ t L t = 1 by [45, Lemma 2.2], and L [t]f L [t] f f . k k · kKω,ν k · k k P Kω,ν k k ≤ k P k · k k ≤ k k Here L [t]f = f is equivalent to t L f = f , since tf = f for f k k 6 k k ω,ν k k 6 k k ∈ L2( ,t). Suppose that P K f = f for some 0 = f L2( ,t). Then it −∞ k t L k k k 6 ∈ −∞ 18 M. SUZUKI implies P Kω,ν f = Kω,νf by Kω,νf = f . Therefore k t L k k L k k L k k k t ∞ Kω,νf(x) 2 dx = Kω,νf(x) 2 dx. | L | | L | Z−∞ Z−∞ Kω,ν Thus L f(x) = 0 for almost every x>t. On the other hand, we have t t Kω,ν ω,ν ω,ν L f(x)= KL (x + y)f(y) dy = KL (x + y)f(y) dy = 0 Z−∞ Z−x 2 Kω,ν for x < t by f L ( ,t). Hence L f has compact support contained in [ t,t]. − ∈ −∞ ω,ν − However, it is impossible for any f = 0 by Lemma 5.2. As the consequence K [t]f < 6 k L k f for every 0 = f L2( ,t). k k 6 ∈ −∞ Kω,ν Finally, we prove iii). As stated in Theorem 2.1, L [t] is a self-adjoint compact Kω,ν operator (because the Hilbert-Schmidt operator is compact). Therefore, L [t] has purely discrete spectrum which has no accumulation points except for 0, and one of Kω,ν Kω,ν Kω,ν L [t] is an eigenvalue of L [t]. However, by ii), every eigenvalue of L [t] has an ±k k ω,ν absolute value less than 1. Hence K [t] < 1. k L k 6. Proof of Theorem 2.4 ω,ν 6.1. Necessity. If we take ν > 1/(ωdL) for each ω > 0, EL satisfies (K1) (K4) by ω,ν ∼ Proposition 4.1. In addition, ΘL is inner in C+ for every (ω,ν) R>0 Z>0 under GRH(L) by Proposition 4.3. Therefore, we obtain (K5) with τ = ∈by Proposition× 5.1. Thus (1), (2) and (3) of Theorem 2.4 hold. Finally, (4) follows from∞ Theorem 2.3. ωn,νn 6.2. Sufficiency. By (1), (2) and (3), we obtain Hamiltonians HL on [0, ) and the t ωn,νn ωn,νn ∞ family of solutions [AL (t, z),BL (t, z)], t 0, of the canonical system on [0, ) ωn,νn ≥ ∞ associated with HL satisfying the initial condition ωn,νn ωn,νn ωn,νn ωn,νn AL (0, z)= AL (z) and BL (0, z)= BL (z). By [45, Proposition 2.4], (4) implies that Eωn,νn (t, z) = Aωn,νn (t, z) iBωn,νn (t, z) be- L L − L longs to HB for every t 0. In particular, ≥ Eωn,νn (0, z)= Aωn,νn (z) iBωn,νn (z)= ξ ( 1 + ω iz)νn L L − L L 2 n − 1 νn 1 νn belongs to HB. That is, ξL( 2 ωn iz) /ξL( 2 + ωn iz) < 1 if (z) > 0. It implies ωn,1 1 | − − − | ℑ1 that EL (z) = ξL( 2 + ωn iz) belongs to HB. In particular, ξL( 2 + ωn iz) has no − 1 − zeros in C+ for every n. This implies that ξL( 2 iz) has no zeros in C+. In fact, if 1 − ωn,1 ξL( 2 iz) has a zero γ = u + iv C+, there exists n such that ωn < v and EL has a zero in− C , since ω 0 and ∈ + n → ξ ( 1 i(u + iv)) = ξ ( 1 + ω i(u + i(v ω ))). L 2 − L 2 n − − n This contradicts Eωn,1 HB for every n. The functional equation implies ξ ( 1 iz) L ∈ L 2 − has no zeros in C . Hence, all zeros of ξ ( 1 iz) are real. − L 2 − 6.3. A variant of Theorem 2.4. By Proposition 4.3 and the above argument, we obtain the following variant of Theorem 2.4. 1 Theorem 6.1. Let L R and 0 <ω0 < 1/2. Then L(s) = 0 for (s) > 2 + ω0 if and only if there exists a sequence∈ S (ω ,ν ) R Z , n 16 , such thatℜ n n ∈ >0 × >0 ≥ (1) ω <ω if m>n and ω ω as n , m n n → 0 →∞ (2) νnωndL > 1, ωn,νn Kωn,νn (3) HL (t) extends to a Hamiltonian on [0, ), that is, det(1 L [t]) = 0 for every t 0, and ∞ ± 6 ωn≥,νn (4) lim J (t; z, z) = 0 for every fixed z C+. t→∞ L ∈ HAMILTONIANS ARISING FROM L-FUNCTIONS IN THE SELBERG CLASS 19 Spectral realization of zeros of ω,ν and ω,ν 7. AL BL ω,ν ω,ν In this part, we mention that the zeros of AL and BL can be regarded as eigenvalues of self-adjoint extensions of a differential operator for ω > 1/2 unconditionally and for 0 <ω 1/2 under GRH(L). ≤ 7.1. Multiplication by the independent variable. For E HB, the de Branges space (E) has the unbounded operator (M, dom(M)) consisting∈ of multiplication by the independentB variable (Mf)(z)= zf(z) endowed with the natural domain dom(M)= f (E) zf(z) (E) . The multiplication operator M is symmetric and closed, sat- isfies{ ∈BM(F ♯)| = (MF∈B)♯ for F} dom(M) and has deficiency indices (1, 1) ([23, Proposition 4.2]). The operator M has no∈ eigenvalues ([23, Corollary 4.3]). In general, dom(M) has codimension at most one. Hereafter, we suppose that dom(M) has codimension zero, that is, dom(M) is dense in (E). Then, all self-adjoint extensions M of M are parametrized by θ [0,π) and theirB spectrum consists of eigenvalues only. θ ∈ The self-adjoint extension Mθ is described as follows. We introduce S (z)= eiθE(z) e−iθE♯(z) θ − for θ [0,π). Then, the domain of M is defined by ∈ θ S (w )F (z) S (z)F (w ) dom(M )= G (z)= θ 0 − θ 0 F (z) (E) θ F z w ∈B 0 − and the operation is defined by MθGF (z)= z GF (z)+ F (w0)Sθ(z), where w is a fixed complex number with S (w ) = 0 and dom(M ) does not depend on 0 θ 0 6 θ the choice of w0. The set S (z) F (z)= θ S (γ) = 0 θ,γ z γ θ − M forms an orthogonal basis of (E), and each Fθ,γ is an eigenfunction of θ with the eigenvalue γ: B MθFθ,γ = γFθ,γ ([23, Proposition 6.1, Theorem 7.3]). We have S (z) = 2i A(z) and S (z)= 2iB(z) π/2 0 − by definition. Therefore, A(z)/(z γ) A(γ) = 0 and B(z)/(z γ) B(γ) = 0 are orthogonal basis of (E) consisting{ − of eigenfunctions| } of M{ and M− , respectively.| } B π/2 0 7.2. Transform to differential operators. Considering the isometric isomorphism of the Hilbert spaces (E) (Θ) L2(0, ), B → K →V0 ⊂ ∞ we define the differential operator D on by V0 −1 −1 D = F MF, dom(D)= F M 1 (dom(M)) = f 0 z(Ff)(z) (Θ) , E { ∈V | ∈ K } d where M 1 is the operator of multiplication by 1/E(z). Then we have (Df)(x)= i f(x) E dx for f C1(R) L2(0, ). All self-adjoint extensions of D are given by ∈ ∩ ∞ −1 −1 Dθ = F MθF, dom(Dθ)= F M 1 (dom(Mθ)), θ [0,π). E ∈ The set −iθ −1 fθ,γ = ie F M 1 Fθ,γ Sθ(γ) = 0 { E | } forms an orthogonal basis of 0 consisting of eigenfunctions fθ,γ of Dθ for eigenvalues γ, −iθ V where ie is the constant for the simplicity of fθ,γ. We have x f (x)= e−iγx 1 (x) e−2iθ K(y)eiγy dy θ,γ [0,∞) − Z0 20 M. SUZUKI −1 by the direct calculation of F M 1 Fθ,γ. This formula suggests that the eigenfunction E f is an adjustment of the “eigenfunction” e−iγx of i d in . θ,γ dx V0 ω,ν Theorem 7.1. Let L R, (ω,ν) R>0 Z>0. Suppose that EL HB so that the ω,ν∈ S ∈ × ω,ν∈ de Branges space (E ) is defined. Then dom(M) is dense in (E ). B L B L Proof. Let E = Eω,ν,Θ= E♯/E. The domain of M is not dense in (E) if and only if L B (γ) < . (7.1) ℑ ∞ γ∈C,XΘ(γ)=0 by [9, Theorem 29] and [1, Theorem A and Corollary 2]. The condition (7.1) means ♯ that the zeros of E appearing in the zeros of Θ (all of them in C+) are finitely many or tend to the real line sufficiently quickly (from the above) if dom(M) is not dense in (E). If the number of such zeros is finite, Θ is a rational function. But it contradicts (4.2)B and (4.3). If E♯ has infinitely many zeros appearing in the zeros of Θ and tend to ♯ the real line, the functional equation of ξL implies that there exists a zero of E above the horizontal line (z)= ω. Hence, (7.1) is impossible. ℑ Theorem 7.1 suggests that the pair ( , D ) is a P´olya-Hilbert space for Aω,ν if θ = π/2 V0 θ L and for Bω,ν if θ = 0. Noting that Aω,1 ξ as ω 0 if ǫ = +1 and Bω,1 iξ as L L → L → L L → L ω,ν(ω) Dω,ν(ω) ω 0 if ǫL = 1, the family of pairs ( 0 , θ ) ω>0 may be considered as a → − { V } ′ perturbation of the “genuine P´olya-Hilbert space” associated with E(z)= ξL(s)+ξL(s), where ν(ω) as ω 0 under (2.8). On the other→∞ hand, → is isomorphic to the quotient space L2(0, )/K(L2( , 0)) V0 ∞ −∞ by [45, Lemma 4.1]. This structure of 0 is similar to Connes’ suggestion for the P´olya- Hilbert space as explained below. V 7.3. Comparison with Connes’ P´olya-Hilbert space. Connes [14] suggests a can- didate of the P´olya-Hilbert space by interpreting the critical zeros of the Riemann zeta function as an absorption spectrum as follows. Let S(R)0 be the subspace of the Schwartz space S(R) consisting of all even functions φ S(R) satisfying φ(0) = (Fφ)(0) = 0. ∈ For a function φ S(R)0, the function Zφ on R+ = (0, ) is defined by (Zφ)(y) = 1/2 ∞ ∈ Z ∞ y n=1 φ(ny). Then φ is of rapid decay as y +0 and y + . In particular, 2 → 2→ ∞ Z(S(R)0) L (R+, dy/y). Then the “orthogonal complement” L (R+, dy/y) Z(S(R)0) P ⊂ −iγ k ⊖ is spanned by generalized eigenfunctions y (log y) , 0 k Major objects of the above naive model of Connes’ idea correspond to objects attached to ( , D ) as follows under the changing of variables y = ex: V0 θ L2(R×, dy/y) L2(0, ) + ⇔ ∞ Z(S(R) ) K(L2( , 0)) 0 ⇔ −∞ Mellin(Z(S(R) )) = ζ( 1 + s)Mellin(S(R) ) F(K(L2( , 0))) = Θ(z)F(L2(0, )) 0 2 0 ⇔ −∞ ∞ d d iy D i , dy ⇔ θ ≈ dx where “Mellin” means the usual Mellin transform and means “is equal up to domain”. ≈ 8. Miscellaneous Remarks (1) Concerning the size of Dirichlet coefficients aL(n), the polynomial bound aL(n) nA for some A 0 is enough to prove Theorems 2.1, 2.2 and 2.3. In other words,| |≪the Ramanujan conjecture≥ (S4) is not necessary to prove these theorems. Therefore, the ω,ν t ω,ν ω,ν method of Sections 3 and 4 about the construction of HL and [AL (t, z),BL (t, z)] can apply to more general L-functions, in particular, to L-functions associated with self- dual irreducible cuspidal automorphic representations of GLn(AQ) with unitary central characters. (2) In contrast with the Ramanujan conjecture (S4), the Euler product (S5) is essential to ω,ν t ω,ν ω,ν the construction of HL and [AL (t, z),BL (t, z)] . In fact, the explicit formula of the ω,ν ω,ν kernel KL coming from (S5) was critical to proved that EL satisfies (K2) and (K3). It ω,ν seems that it is not easy even to prove that KL is a function if we do not have (S5). It ω,ν t ω,ν ω,ν is an interesting problem to extend the construction of HL and [AL (t, z),BL (t, z)] to the class of L-data which is an axiomatic framework for L-functions introduced by A. Booker [3]. Superficially, Booker’s L-datum does not require the Euler product, but it is based on the Weil explicit formula of L-functions in the Selberg class. Roughly, the Weil explicit formula is a result of (S3) and (S5), but the theory of L-data suggests that the Weil explicit formula is more essential than (S3) and (S5). (3) By the Euler product (S5), L is expressed as a product of local p-factors L , ∈ S p and often, there exists polynomial Fp of degree at most dL for each prime p such that −s Lp(s) = 1/Fp(p ). The Ramanujan conjecture (S4) is understood as an analogue of −s −s the Riemann hypothesis for Fp(p ). The Hamiltonian HL,p attached to Fp(p ) was constructed in [44] by using a way analogous to the method in Section 3 if Fp is a real self-reciprocal polynomial (for details, see [44, Section 1, Section 7.6]). It is an interesting ω,ν problem to find a relation among the perturbation family of global Hamiltonians HL , the family of local Hamiltonians HL,p and the conjectural Hamiltonian HL corresponding ′ to E(z)= ξL(s)+ ξL(s). (4) The method of [44] for the construction of H and t[A(t, z),B(t, z)] for exponen- ω,ν tial polynomials E is useful to observe HL by numerical calculation of computer for concrete given L R, because an entire function satisfies (K1) is approximated by exponential polynomials∈ S by approximating the Fourier integral by Riemann sums (cf. the final part of the introduction of [44]). ω,ν (5) A sharp estimate of KL (x) for large x > 0 is not necessary to prove the main ω,ν results of this paper. In fact, we do not know the role of the behavior of KL (x) at x =+ in the equivalent condition of Theorem 2.4. ∞ (6) If we replace the condition (4) of Theorem 2.4 by (4’) Eωn,νn (0) = 0 and lim (Aωn,νn (t, z),Bωn,νn (t, z)) = (Eωn,νn (0), 0), L 6 t→∞ L L L we obtain a sufficient condition for GRH(L), since (4)’ implies (4). It is ideal if this is also a necessary condition, but we have no plausible evidence to support the necessity 22 M. SUZUKI ω,ν of (4)’. On the contrary, it is not clear whether limt→+∞ AL (t, z) defines a functions ω,ν of z contrast with the fact limt→+∞ BL (t, z) = 0 under ω > 1/2 or GRH(L) (see [45, Section 5.4]). The limit behavior may be related to the arithmetic properties of L(s) in ω,ν a deep level, because we need information of all qL (n)’s to understand it differ from ω,ν Kω,ν the situation that we need only finitely many qL (n)’s to understand L [t] for a finite range of t R. We do not touch this problem further in this paper. ∈ References [1] A. 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