Amplified Fourth Moment of the Riemann Zeta-Function and Applications

TL;DR

This paper derives an asymptotic formula for the fourth moment of the Riemann zeta-function multiplied by an amplifier, advancing understanding of zero gaps and moments in number theory.

Contribution

It introduces a new asymptotic formula for the zeta-function's fourth moment with an amplifier, extending previous results to more general Dirichlet polynomial applications.

Findings

Asymptotic formula for the fourth moment with an amplifier

Applications to gaps between zeros of ζ(s)

Lower bounds for moments of ζ(s)

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 $1/\zeta^r(s)$ (a mollifier) or $\zeta(s)^r$ (an amplifier) for some $r>0$. Previous known results include the mean value of the fourth power of $\zeta(s)$ 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 $\zeta(s)$ and lower bounds for moments.