On Li's criterion for the Riemann hypothesis for the Selberg class

On Li’s coefficients for the Rankin–Selberg L-functions

“…The second author [28] has defined the generalized Li coefficient λ n (F ) for a function F belonging to a certain subclass of the extended Selberg class (that contains the Selberg class), proved that the generalized Riemann hypothesis in this case is equivalent to the non-negativity of numbers Re λ n (F ), for all n ∈ N and obtained two arithmetic expressions for the generalized Li coefficients. In [22] the authors have obtained the full asymptotic expansion of the archimedean part of λ n (F ) in terms of n log n, n, n 0 and odd negative powers of n, as well as the approximation of the finite part of λ n (F ) in terms of the incomplete Li coefficient up to a height √ n. In this paper we study the generalized Li coefficients associated with L-functions attached to the Rankin-Selberg convolution of two unitary cuspidal automorphic representations π and π of GL m (A F ) and GL m (A F ).…”

On the asymptotic criterion for the zero-free regions of certain $L$-functions

“…We will refer to this criterion as the asymptotic Li criterion. These results were generalized for the class S ♯♭ in [11].…”

Euler-Stieltjes constants for the Rankin-Selberg L-function and weighted Selberg orthogonality

“…where Λ F (n) is generalized von Mangoldt function defined in terms of the coefficients in the Euler product of F and m F is the order of (eventual) pole of F ∈ S ♭ at s = 1, see [36,Corollary 6.3.]. Let E be a Galois extension of Q of a finite degree and let π and π ′ be irreducible unitary cuspidal representations of GL m (E A ) and GL m ′ (E A ), respectively.…”

“…In this setting, it is possible to normalize representations π and π ′ so that the "normalized" Rankin-Selberg L−function possesses a pole at s = 1, hence computation of coefficients in the Laurent series expansion of its logarithmic derivative at s = 1 reduces to a slight modification of the case treated in [36].…”