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

Zhaoyan Chen

TL;DR

The article establishes asymptotic formulas for mixed moments of the Riemann zeta function and a Hecke $L$-function attached to a Hecke cusp form on the critical line, including a result for the product with $|ζ|^{2}$. The approach hinges on exact approximate functional equations, careful analysis of Dirichlet-series sums, and stationary-phase methods to handle oscillatory integrals, yielding a main term of the form $\frac{c}{2} L(1,f) T \log T + c_f T$ with a power-saving error. The work extends classical moment results to the joint setting with automorphic $L$-functions and derives short-interval corollaries by leveraging known moment bounds and smoothing arguments. This provides precise mean-value information for mixed zeta–$L$-function moments, with implications for understanding correlations between $ζ(s)$ and automorphic $L$-functions. The results unify analytic techniques for moments, approximate functional equations, and oscillatory integral estimates in the $GL(2)$ automorphic setting.

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 $ζ(s)$ and $L(s,f)$ on the critical line. Similarly, we prove an asymptotic formula for the mixed moment of the product of $|ζ^{2}(s)|$ and $L(s,f)$ on the critical line.

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

TL;DR

The article establishes asymptotic formulas for mixed moments of the Riemann zeta function and a Hecke -function attached to a Hecke cusp form on the critical line, including a result for the product with . The approach hinges on exact approximate functional equations, careful analysis of Dirichlet-series sums, and stationary-phase methods to handle oscillatory integrals, yielding a main term of the form with a power-saving error. The work extends classical moment results to the joint setting with automorphic -functions and derives short-interval corollaries by leveraging known moment bounds and smoothing arguments. This provides precise mean-value information for mixed zeta–-function moments, with implications for understanding correlations between and automorphic -functions. The results unify analytic techniques for moments, approximate functional equations, and oscillatory integral estimates in the automorphic setting.

Abstract

Let be a Hecke cusp form for . We prove an asymptotic formula for the mixed moment of the product of and on the critical line. Similarly, we prove an asymptotic formula for the mixed moment of the product of and on the critical line.
Paper Structure (9 sections, 9 theorems, 101 equations)

This paper contains 9 sections, 9 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximate functional equations
  4. Sums of Fourier coefficients
  5. Estimation of integrals
  6. Proof of Theorem \ref{['t1']}
  7. Proof of Corollaries
  8. Proof of Corollary \ref{['c3']}
  9. Proof of Corollary \ref{['c4']}

Key Result

Theorem 1.1

Let $f$ be a Hecke cusp form (holomorphic or Maass) for $SL(2,\mathbb{Z})$. Suppose $V(x)$ is a smooth function with support contained in $[1,2]$, further satisfying the condition for $i\geq0$ with $\Delta\leq T^{1/2-\varepsilon}$. For any $\varepsilon>0$, then we have that where $c=\int_{\mathbb{R}}V(\xi)\mathrm{~d} \xi$ and the constant $c_{f}$ is given by

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 10 more