Amplified Fourth Moment of the Riemann Zeta-Function and Applications

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.