On the Discretized Li Coefficients for a Certain Class of L-Functions.

The τ -Keiper/Li coefficients attached to a function F are closely related to its zero-free regions. However, the absence of a closed formula for calculating these coefficients makes them challenging to use. Motivated by Voros approach, we introduce the discretized τ -Keiper/Li coefficients. A finite sum representation derived for these coefficients is useful for numerical calculations. Representation in terms of zeros of the corresponding function is basis for analytic considerations. We prove that the violation of τ / 2 -generalized Riemann hypothesis implies oscillations of corresponding discretized τ -Li coefficients with power-growing amplitudes. Results are obtained for the class S ♯ ♭ (σ 0 , σ 1) , which contains the Selberg class, the class of all automorphic L-functions, the Rankin–Selberg L-functions, as well as products of suitable shifts of those functions. [ABSTRACT FROM AUTHOR]