A note on the mean square of the Riemann zeta-function

TL;DR

The paper investigates the mean-square of the Riemann zeta-function in the presence of a finite Dirichlet polynomial $A(s)$ by introducing a new weight function $\omega(t,T_1,T_2)$ to refine the standard mean-square expansion. It builds on the approach of Balasubramanian, Conrey, and Heath-Brown, employing Mellin-transform techniques, contour integration, and residue calculus to express the main term as a double Dirichlet-sum with a logarithmic factor, and to rigorously bound the error via Lemmas 2.4–2.6. The main result states that $I = T\sum_{h,k\le M}\frac{a(h)}{h}\frac{\overline{a(k)}}{k}(h,k)\left(\log\frac{T(h,k)^2}{2\pi hk}+2\gamma-1\right) + \tilde{\mathscr{E}}$ with $\tilde{\mathscr{E}} \ll V T^{-\eta+\varepsilon} + o(T)$ and, since $V \ll M^{2+2\varepsilon+2\eta}$, the combined error is $\tilde{\mathscr{E}}, \mathscr{E} \ll M^2 T^{\varepsilon}$. The work clarifies the role of the auxiliary weight and connects the constant term to Ingham's $2\gamma$, yielding a sharp mean-square formula with a power-saving error.

Abstract

In this paper, we will give a new proof for a known result of the mean square of Riemann zeta-function.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 2.6