Amplified Fourth Moment of the Riemann Zeta-Function and Applications

TL;DR

The paper proves an asymptotic formula for the fourth moment of the Riemann zeta-function multiplied by the fourth power of an amplifier built from divisor-type coefficients, valid for amplifier length $y=T^{\vartheta}$ with $\vartheta<1/8$. The core method combines the Bettin–Bui–Li–Radziwiłł twisted moment framework with Mellin-type expansions and contour analysis to extract the main term, yielding an explicit leading term involving $a_{2r+2}$ and a multidimensional integral $c(\underline{α},\underline{β})$, plus a controlled error. This result enables applications to the zero-distribution of $\zeta(s)$, notably improving unconditional lower bounds for gaps between consecutive zeros by coupling the amplified moment with a Wirtinger-type inequality; the paper establishes $\Lambda>2.64$ for the normalized gaps. The work thus advances both moment calculations with amplifiers and the unconditional study of zero spacings, with potential further improvements via larger amplifiers or higher-degree mollifiers. All formulas are presented with explicit constants and integrals, and the techniques unify mean-value theorems with nonlinear optimization over amplifier parameters.

Abstract

The twisted fourth moment of the Riemann zeta-function was established by Hughes and Young [J. Reine Angew. Math. 641 (2010), 203--236] and later improved by Bettin, Bui, Li and Radziwill [J. Eur. Math. Soc. (JEMS) 22 (2020), 3953--3980]. In applications one would often like to take the Dirichlet polynomial to mimic either $1/ζ^r(s)$ (a mollifier) or $ζ(s)^r$ (an amplifier) for some $r>0$. Previous known results include the mean value of the fourth power of $ζ(s)$ times the square or the fourth power of a mollifier, or the square of an amplifier. In this paper we obtain the asymptotic formula for the fourth moment of the Riemann zeta-function times the fourth power of an amplifier. This has various applications to the theory of the Riemann zeta-function, e.g. gaps between zeros of $ζ(s)$ and lower bounds for moments.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1: Bettin, Bui, Li and Radziwiłł