Integral moment of the Riemann zeta function and Hecke $L$ functions

TL;DR

This paper derives asymptotic formulas for mixed moments involving the Riemann zeta function and Hecke L-functions on the critical line, advancing understanding of their joint behavior in analytic number theory.

Contribution

It provides new asymptotic formulas for the mixed moments of ta and Hecke L-functions, a novel result in the study of their joint distribution.

Findings

Asymptotic formulas for mixed moments of ta and L-functions established.

Results enhance understanding of the joint behavior of ta and Hecke L-functions.

Advances in analytic techniques for evaluating moments on the critical line.

Abstract

Let $f$ be a Hecke cusp form for $SL(2,\mathbb{Z})$. We prove an asymptotic formula for the mixed moment of the product of $\zeta(s)$ and $L(s,f)$ on the critical line. Similarly, we prove an asymptotic formula for the mixed moment of the product of $|\zeta^{2}(s)|$ and $L(s,f)$ on the critical line.