Asymptotic Relations Between Interpolation Differences and Zeta Functions

TL;DR

This paper explores the asymptotic relationships between various zeta functions and interpolation differences of certain functions, providing new criteria for locating zeros within the critical strip.

Contribution

It introduces novel asymptotic relations and criteria for zeros of zeta functions based on interpolation differences and integrability conditions.

Findings

Established asymptotic relations between zeta functions and interpolation differences.

Derived new criteria for zeros of zeta functions in the critical strip.

Connected integrability of interpolation differences with zero distribution.

Abstract

Asymptotic relations between zeta functions (such as, $\zeta(s),\,\beta(s)$, and other Dirichlet $L$-functions) and interpolation differences of functions like $\vert y\vert^s$ and their interpolating entire functions of exponential type $1$ are discussed. New criteria for zeros of the zeta functions in the critical strip in terms of integrability of the interpolation differences are obtained as well.