On the \(\tau \)-Li coefficients for automorphic \(L\)-functions (Q1683399)

The Li coefficients in the case of the Riemann \(\zeta\)-function \(\zeta(s)\) are defined by \textit{X.-J. Li} [J. Number Theory 65, No. 2, 325--333 (1997; Zbl 0884.11036)] as \[ \lambda_n={\sum_{\rho}}^*\left[ 1-\left(1-\frac{1}{\rho}\right)^n \right], \] where \(n\) is a positive integer, and the sum is over non-trivial zeros \(\rho\) of \(\zeta(s)\), counted with multiplicities. The \(*\) always denotes the limit as \(|Im(\rho)|\) tends to infinity, that is, the limit as \(T\to\infty\) of the sum taken over all zeros \(\rho\) such that \(|Im(\rho)|<T\). The Li criterion relates the Riemann hypothesis to the non-negativity of the Li coefficients \(\lambda_n\). The Li coefficients are generalized to the case of other Dirichlet series, and the Li criterion is shown to hold for the Dirichlet series in the Selberg class [\textit{S. Omar} and the author, J. Number Theory 125, No. 1, 50--58 (2007; Zbl 1137.11060); ibid. 130, No. 4, 1109--1114 (2010; Zbl 1188.11045)]. In the case of principal automorphic \(L\)-functions for \(\mathrm{GL}_N\) over \(\mathbb{Q}\), the Li coefficients were defined, and the Li criterion for the generalized Riemann hypothesis was shown by \textit{J. Lagarias} [Ann. Inst. Fourier 57, No. 5, 1689--1740 (2007; Zbl 1216.11078)]. Extending the Li coefficients in a certain way, leads to the Li criterion for the zero-free regions in the critical strip for Dirichlet series and automorphic \(L\)-functions. In the present paper, the author extends the Lie coefficients of the principal automorphic \(L\)-function attached to a cuspidal automorphic representation \(\pi\) of \(GL_N(\mathbb{A}_\mathbb{Q})\), where \(\mathbb{A}_\mathbb{Q}\) is the ring of adèles of \(\mathbb{Q}\). Given \(\tau\geq 1\) and a positive integer \(n\), the extended Li coefficient, or the \(\tau\)-Li coefficient, for \(\pi\) is defined as \[ \lambda_n(\pi,\tau)= {{\sum}^*_{\rho\in Z(\pi)}}\left[ 1-\left(1-\frac{\rho}{\rho-\tau}\right)^n \right], \] where \(Z(\pi)\) is the multiset of zeros, counted with multiplicities, of appropriately normalized complete principal automorphic \(L\)-function \(\Lambda(s,\pi)\) attached to \(\pi\). This definition is similar to that of \textit{P. Freitas} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 399--414 (2006; Zbl 1102.11046)]. The first main result of the present paper is the generalization of the Li criterion for zero-free regions of the automorphic \(L\)-function \(\Lambda(s,\pi)\) inside the critical strip. More precisely, the existence of the zero-free vertical strips of the form \(\tau /2 < \mathrm{Re}(\rho)<1\) is related to the non-negativity of the \(\tau\)-Li coefficients \(\lambda_n(\pi,\tau)\) for all positive integers \(n\). This is the extension to the general class of principal automorphic \(L\)-functions of the result of \textit{P. Freitas} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 399--414 (2006; Zbl 1102.11046)]. Moreover, an arithmetical and an asymptotical formula for the \(\tau\)-Li coefficients \(\lambda_n(\pi,\tau)\) is obtained. Finally, using the argument of \textit{A. Odžak} and \textit{L. Smajlović} [Int. J. Number Theory 7, No. 3, 771--792 (2011; Zbl 1312.11074)], an entire function of order one and finite exponential type which interpolates the Archimedean contribution and the finite contribution of \(\tau\)-Li coefficients \(\lambda_n(\pi,\tau)\) at positive integers is determined.