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On the modified Li criterion for a certain class of \(L\)-functions. (English) Zbl 1347.11063
The authors consider a class \(\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)\) of \(L\)-functions that contains the Selberg class, the class of all automorphic \(L\)-functions and the Rankin-Selberg \(L\)-functions, as well as products of suitable shifts of those functions. They prove the generalized Li criterion for zero-free regions of functions belonging to the class \(\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)\), derive an arithmetic formula for \(\tau\)-Li coefficients and conduct numerical investigation of \(\tau\)-Li coefficients for a certain product of shifts of the Riemann zeta function.
More precisely, for real numbers \(\sigma_0\) and \(\sigma_1\) such that \(\sigma_0 \geq \sigma_1>0\), the class \(\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)\) is the class of functions \(F\) satisfying the following four axioms:
\(\bullet\)
(Dirichlet series) The function \(F\) possesses a Dirichlet series representation that converges absolutely for \(\mathrm{Re} (s) >\sigma_0.\)
\(\bullet\)
(Analytic continuation) There exist finitely many non-negative integers \(m_1, \dots, m_N\) and complex numbers \(s_1, \dots, s_N\) such that the function \(\prod\limits_{i=1}^{N} (s-s_i)^{m_i}F(s)\) is an entire function of finite order.
\(\bullet\)
(Functional equation) The function \(F\) satisfies the functional equation \( \xi_{F}(s)=\omega \overline{\xi_{F}(\sigma_1-\bar{s})}, \) where the completed function \(\xi_F\) is defined as \[ \begin{split} \xi _{F}(s)=&F(s) Q_{F}^{s}\prod_{j=1}^{r}\Gamma (\lambda _{j}s+\mu _{j})\prod\limits_{i=1}^{2M + \delta(\sigma_1)} (s-s_i)^{m_i} \\ &\prod_{i=2M+1 + \delta(\sigma_1)}^{N}(s-s_i)^{m_i} (\sigma_1 - s-\overline{s_i})^{m_i}, \end{split} \] where \(\left| \omega \right| =1\), \(Q_{F}>0\), \(r\geq 0\), \(\lambda _{j}>0\), \(\mu_j\in\mathbb C\), \( j=1,\ldots ,r\), and \(\Gamma\) is the Euler Gamma function. It is assumed that the poles of \(F\) are arranged so that the first \(2M+\delta(\sigma_1)\) poles (\(0\leq 2M + \delta(\sigma_1) \leq N\)) are such that \(s_{2j-1} + \overline{s}_{2j} = \sigma_1\), for \(j=1,\ldots, M\), and \(\delta(\sigma_1) = 1\) if \(\sigma_1/2\) is a pole of \(F\) in which case \(s_{2M+\delta(\sigma_1)} = \sigma_1/2\); otherwise \(\delta(\sigma_1)=0\).
\(\bullet\)
(Euler sum) The logarithmic derivative of the function \(F\) possesses a Dirichlet series representation
converging absolutely for \(\mathrm{Re} (s)>\sigma_0\).
\smallskip
The non-trivial zeros of \(F\) are defined to be the zeros of the completed function \(\xi_F\). The set of non-trivial zeros of \(F(s)\) is denoted by \(Z(F)\). By the functional equation and the Euler sum, all those zeros lie in the critical strip \(\sigma_1-\sigma_0\leq \mathrm{Re} (s) \leq \sigma_0\). Let \(\tau\in[\sigma_1,+\infty)\). For an arbitrary positive integer \(n\), the \(n\)th \(\tau\)-Li coefficient associated to the \(F\in\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)\) is defined as \[ \lambda_{F}(n,\tau)=\left.\sum_{\rho\in Z(F)}\right.\left(1-\left(\frac{\rho}{\rho-\tau}\right)^n\right), \] where the sum is taken in the sense of the limit \(\lim\limits_{T \to \infty} \sum\limits_{|\mathrm{Im}(\rho)| \leq T}\). The main result of the paper is the following Li-type criterion. Let \(0,\tau\notin Z(F)\). The next two statements are equivalent
(i)
\(\sigma_1-\frac{\tau}{2}\leq \mathrm{Re} (\rho)\leq\frac{\tau}{2}\) for every \(\rho\in Z(F)\),
(ii)
\(\mathrm{Re} (\lambda_F(n,\tau))\geq 0\) for every positive integer \(n\).

MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11M41 Other Dirichlet series and zeta functions
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