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

Zhaoyan Chen

TL;DR

This work establishes an asymptotic formula for the mixed moment $\int_{-\infty}^{\infty} V(t/T)\, L(\tfrac{1}{2}+it,f)\, \zeta(\tfrac{1}{2}-it)^2\, dt$ on the critical line, for a Hecke cusp form $f$ (holomorphic or Maass). The author combines approximate functional equations, Voronoi summation, Mellin transforms, and stationary-phase analysis to isolate a main diagonal contribution $2cT\frac{L(1,f)^2}{\zeta(2)}$ (with $c=\int V$) and to bound the remaining terms by $O(T^{1/2+\varepsilon})$. The approach handles both the $t$-aspect and oscillatory test functions, enabling unweighted and smoothed formulations and yielding corollaries about related moments and mean-square bounds. The results extend prior mixed-moment studies by removing absolute values and providing uniform treatment across holomorphic and Maass cases, with implications for the understanding of moments of automorphic $L$-functions.Overall, the paper advances the analytic theory of mixed moments on the critical line by delivering a sharp asymptotic formula and robust error control using a synthesis of approximate functional equations, Voronoi theory, Mellin techniques, and stationary-phase analysis.

Abstract

In this paper, let $f$ be a Hecke cusp form for $SL(2,\mathbb{Z})$. We establish an asymptotic formula for the mixed moment of $ζ^{2}(s)$ and $L(s,f)$ on the critical line, valid for both holomorphic and Maass forms.

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

TL;DR

This work establishes an asymptotic formula for the mixed moment on the critical line, for a Hecke cusp form (holomorphic or Maass). The author combines approximate functional equations, Voronoi summation, Mellin transforms, and stationary-phase analysis to isolate a main diagonal contribution (with ) and to bound the remaining terms by . The approach handles both the -aspect and oscillatory test functions, enabling unweighted and smoothed formulations and yielding corollaries about related moments and mean-square bounds. The results extend prior mixed-moment studies by removing absolute values and providing uniform treatment across holomorphic and Maass cases, with implications for the understanding of moments of automorphic -functions.Overall, the paper advances the analytic theory of mixed moments on the critical line by delivering a sharp asymptotic formula and robust error control using a synthesis of approximate functional equations, Voronoi theory, Mellin techniques, and stationary-phase analysis.

Abstract

In this paper, let be a Hecke cusp form for . We establish an asymptotic formula for the mixed moment of and on the critical line, valid for both holomorphic and Maass forms.
Paper Structure (10 sections, 16 theorems, 103 equations)

This paper contains 10 sections, 16 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximate functional equations
  4. Sums of Fourier coefficients
  5. Estimation of integrals
  6. Averages of Kloosterman sums
  7. Mellin transform and Proof of Theorem \ref{['t1']}
  8. Mellin transform
  9. Proof of Theorem \ref{['t1']}
  10. Proof of Corollaries

Key Result

Theorem 1.1

Let $f$ be a Hecke cusp form (either holomorphic or Maass) for $SL(2,\mathbb{Z})$. Suppose $V$ is a smooth function supported on $[1,2]$ satisfying for $i\geq 0$ with $\Delta\ll T^{\varepsilon}$. For any $\varepsilon > 0$, we have that where $c = \int_{\mathbb{R}} V(\xi)\mathrm{~d} \xi$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 18 more