On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients

Abstract

The Li coefficients λF (n) of a zeta or L-function F provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the τ -Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport-Heilbronn zeta function. The behavior of the τ -Li coefficients varies depending on whether the function in question has any zeros in the half-plane Re(z) > τ /2. We investigate analytically and numerically the behavior of these coefficients for such functions in both the n and τ aspects.

Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic

“…More recently, Arb has been used to study generalizations of the Keiper-Li coefficients [24]. Related to this example, Matiyasevich and Beliakov have also performed investigations of Dirichlet L-functions that involved using Arb to locate zeros to very high precision [25], [26].…”

Women in Numbers Europe

“…1, for different values of 1, depending on the real parts of shifts. In the paper Bucur et al (2015), we define generalized -Li coefficients for functions F in the class S ] R that consists of all functions F belonging to the extended Selberg class S ] , introduced in Kaczorowski and Perelli (1999) with the property that is a zero of F if and only if 1 is. Functions from the class S ] R do not necessarily satisfy the generalized Riemann hypothesis.…”