On the Second Moment of Twisted Higher Degree $L$-functions

TL;DR

This paper develops a method to bound the second moment of twisted higher degree $L$-functions in the $q$-aspect, assuming key conjectures, achieving a log-saving upper bound that improves understanding of these functions' behavior.

Contribution

It introduces a general approach to obtain log-saving bounds for the second moment of standard twisted higher degree $L$-functions under certain conjectural hypotheses.

Findings

Achieves a bound of rac{q^{d/2}}{ ext{log}^{ ext{eta}} q} for the second moment.

Relies on the Ramanujan conjecture, zero density estimates, and subconvexity bounds.

Provides a framework applicable to $L$-functions of degree $d \\geq 3$.

Abstract

Assuming the Ramanujan conjecture, the zero density estimate and some subconvexity type bound, we describe a general method to obtain the log-saving upper bound for the second moment of standard twisted higher degree $L$-function in the $q$-aspect. Specifically, let $L(s, F)$ be a standard $L$-function of degree $d\geq3$. Under these foundational hypotheses. the bound \[ \sideset{}{^*}{\sum}_{\chi \pmod q}\Big|L\big(\frac{1}{2}, F\times \chi \big)\Big |^2\ll_{F,\eta} \frac{q^{\frac{d}{2}}}{\log^{\eta}q} \] holds for some small $\eta>0$