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Screw functions of Dirichlet series in the extended Selberg class

We introduce screw functions for Dirichlet series in the extended Selberg class. Then we prove that the Grand Riemann Hypothesis for a member of the extended Selberg class is equivalent to the nonpositivity of the corresponding screw function.


Introduction
In [8], we introduced a screw function corresponding to the Riemann zeta-function whose nonpositivity of values is equivalent to the Riemann hypothesis for the Riemann zeta-function. As a large class of functions similar to the Riemann zeta-function, the Selberg class S and the extended Selberg class S ♯ are well-known and important in number theory. The Riemann zeta-function, Dirichlet L-functions, and L-functions of holomorphic cusp forms are typical members of S (cf. [6]). Therefore, it is naturally expected that a member F of the extended Selberg class also corresponds to a screw function similarly to the Riemann zeta-function. This paper implements that expectation.
Let F be a member of the extended Selberg class S ♯ . The nontrivial zeros of F (s) mean the zeros of the entire function ξ F (s), which is the Selberg class analogue of the Riemann xi-function defined in (2.3) below. We call γ ∈ C a nontrivial zero of F (1/2−iz) if 1/2−iγ is a nontrivial zero of F (s). The Grand Riemann Hypothesis (GRH, for short) for F is the assertion that all nontrivial zeros of F (s) lie on the line ℜ(s) = 1/2, which is equivalent that all nontrivial zeros of F (1/2 − iz) are real. We define the function g F (t) on the real line by for nonnegative t and set g F (t) = g F (−t) for negative t, where Γ ♯ F is the set of all nontrivial zeros of F (1/2 − iz), m γ is the multiplicity of γ ∈ Γ ♮ F , and m 0 = 0 if 0 ∈ Γ ♯ F . For ease of introduction, definition (1.1) refers to [8,Proposition 2.3] rather than definition [8, (1.1)] of Φ(t) = −g(t), because we need the semi-extended Selberg class S ♯♭ introduced by Smajlović [7] (and reviewed in Section 2) to define g F (t) without using zeros (cf. Theorem 4.1 below). The difference from the case of the Riemann zeta-function is the existence of the term −m 0 t 2 /2 corresponding to γ = 0 and the lack of symmetry γ → −γ in the sum over zeros. We call g F (t) the screw function of F because it is a screw function in the sense of [4] assuming the GRH for F is true (see Section 5 below). The main result of the present paper is the following generalization of [8,Theorem 1.2].
According to the proof of Theorem 1.1 in Section 3 below, if F ∈ S, ℜ(−g F (t)) ≥ 0 for all t ≥ t 0 , and s = 1 is neither a zero nor a pole of F (s), then there exists δ F > 0 such that F (s) has no zeros in the right-half plane ℜ(s) > 1 − δ F , since F ∈ S has no zeros in ℜ(s) > 1 by the Euler product. On the other hand, F ∈ S ♯ \ S may have a real zero greater than one.
In the following, we review the Selberg class in Section 2, and prove Theorem 1.1 in Section 3. Then we show that the screw function g F allows a representation without zeros if we restrict F to the semi-extended Selberg class S ♯♭ in Section 4. Finally, we comment mainly on generalizations of other results in [8] in Section 5.

The Selberg class and explicit formulas
The Selberg class S introduced by A. Selberg in 1992 consists of the Dirichlet series where b F (n) = 0 unless n = p m with m ≥ 1, and b F (n) ≪ n θ for some θ < 1/2. From (S3) and (S5), F ∈ S has no zeros outside the critical strip 0 ≤ ℜ(s) ≤ 1 except for zeros in the left half-plane ℜ(s) ≤ 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 F ≡ 1 and coincide with the zeros of the entire function ξ F (s) of (2.3). The extended Selberg class S ♯ is the class of functions satisfying (S1)-(S3) above. Note that the data (ω, Q, r, λ j , µ j ) in (S3) are not uniquely defined by F ∈ S ♯ . For (2.4) The survey [6] and its sequel are good introductions to the theory of the Selberg class and the extended Selberg class.
To represent the screw function g F (t) without using zeros, we recall the subclass S ♯♭ introduced in [7, Section 2] that consists of F ∈ S ♯ satisfying the Euler sum condition: (S5') The logarithmic derivative of F possesses a Dirichlet series representation converging absolutely for ℜ(s) > 1. The subclass S ♯♭ , which we call the semi-extended Selberg class, contains the Selberg class S ([7, Theorem 2.1]). The Euler sum condition (S5') is satisfied if log F (s) has a Dirichlet series with suitable convergence. For a condition of Dirichlet series representation in log F (s), see [1,Theorem 11.14] for example, but see also [5,Satz 12] for the convergence of the Dirichlet series. s). Therefore, assuming the GRH for F is true, The values of −g F (t) are clearly nonnegative from the right-hand side.
To prove the converse claim, we first show that holds when ℑ(z) > 1/2. For F ∈ S ♯ , the function ξ F (s) is an entire function of order one ([7, Lemma 3.3 and a note below the proof]). Therefore, the Hadamard factorization theorem implies that ξ F (1/2 − iz) has the product formula where the sum on the right-hand side converges absolutely and uniformly on every compact subset of C \ Γ ♮ F . For each term on the right-hand side, we have Hence we obtain (3.1). By (2.2) and (2.4), the mapping γ → −γ defines a bijection from Γ F to Γ F * with multiplicity, and B F = −B F * (⊂ R). Applying these facts to (1.1), (3.4) Therefore, we obtain when ℑ(z) > 1/2. Let σ 0 := max{σ ∈ R | ξ F (σ) = 0}. This number is the same for F * by the functional equation (2.2) . If ℜ(−g F (t)) is nonnegative for t ≥ t 0 , then the integral on the left-hand side converges when ℑ(z) > max(σ 0 − 1/2, 0) by [9, Theorem 5b of Chapter II]. Hence ξ F (s) and ξ F * (s) have no zeros in the right-half plane ℜ(s) > max(σ 0 − 1/2, 0). In particular, the GRH for both F and F * are true if ξ F (s) or ξ F * (s) has no real zeros except for the possible zero at s = 1/2.

Zero-free formulas of screw functions
For functions F in the semi-extended Selberg class S ♯♭ , we obtain the following zerofree formulas of corresponding screw functions.
Theorem 4.1. Let F ∈ S ♯♭ and let g F (t) be the screw function of (1.1). We chose the data ω, Q, r, λ j , and µ j satisfying (S3) and define Proof. For the equality Φ F (t) = −g F (t), it suffices to prove that holds when ℑ(z) > 1/2 by (3.1) and the uniqueness of the inverse Fourier transform. Taking the logarithmic derivative of (2.3), On the other hand, we have if ℑ(z) > 1/2 by (S5'). Hence, (4.2) is proved if it is shown that holds for λ > 0, µ ∈ C with ℜ(µ) ≥ 0, and ℑ(z) > 0. We have for a complex number β by direct and simple calculation. Therefore, Using this and the series expansion Φ(z, s, a) = ∞ n=0 z n (n + a) −s , the left-hand side of (4.5) is calculated as The right-hand side is equal to the right-hand side of (4.5) by the well-known series expansion where γ 0 is the Euler-Mascheroni constant. Hence, we complete the proof of (4.2). On the other hand, it is known that the γ-factor γ(s) = Q s r j=1 Γ(λ j s + µ j ) is uniquely determined up to a constant multiple ([6, Theorem 4.1]). Therefore, the logarithmic derivative (γ ′ /γ)(s) is uniquely determined from F . Hence the values of Φ F (t) are uniquely determined from F by (4.2).

Complements
Finally, we make several comments on generalizations of other results in [8]. (3.2), where the sum is taken over |γ| ≤ T and then letting T → ∞ for the convergence. By the functional equation (2.2), Γ F is closed under the complex conjugation, thus γ γ −1 is real. Therefore, assuming the GRH for F ,

The theory of screw functions is somewhat simpler for real-valued function ([4]).
The screw function g F for F ∈ S ♯ is real-valued if F = F * by (3.4). In general F ∈ S ♯ does not satisfy F = F * , but since S ♯ is a multiplicative monoid, G := F F * belongs to S ♯ and satisfies G = G * . Since the GRH for F is equivalent to that for the product F F * by (2.2), it is sufficient to consider F F * to study the GRH. In other words, we only need to study F ∈ S ♯ satisfying F = F * .
5.3. The generalization of [8, Theorem 1.1] to F ∈ S ♯ follows immediately from the integral representation (3.1), but generalizing [8, Theorem 1.3] requires an additional assumption to prove the convergence of moments µ F,n = ∞ 0 e −t/2 (−g F (t)) t n dt for all nonnegative integers n. If we suppose that F ∈ S and it has a zero-free region ℜ(s) > 1 − C(log(3 + |ℑ(s)|)) −1 for some C > 0, then we obtain (F ′ /F )(σ + it) ≪ log(3 + |t|) in the zero-free region by standard arguments as in [3,Section 5.6]. With this estimate, the argument in [8,Section 4] works and the convergence of the moments µ F,n is obtained. 5.4. In case F ∈ S ♯♭ , the GRH for F is true if and only if for all smooth and compactly supported functions φ on R by Smajlović's explicit formula [7, Theorem 3.1], the generalization of Li's criterion [7,Theorem 4.3], and argument of the proof of [2, Theorem 1]. Therefore, the generalization of [8,Theorem 1.4] for F ∈ S ♯♭ holds by replacing g with g F . The result [8,Theorem 1.5] is also expected to be generalized to the semi-extended Selberg class F ∈ S ♯♭ , but detailed calculations for confirmation are left for future work.