On the second moment of twisted higher degree $L$-functions
Volume 222 / 2026
Acta Arithmetica 222 (2026), 107-135
MSC: Primary 11F66; Secondary 11N75
DOI: 10.4064/aa250119-11-8
Published online: 28 December 2025
Abstract
Assuming the Ramanujan conjecture, the zero density estimate and some subconvexity type bound, we describe a general method of obtaining the log-saving upper bound for the second moment of a standard twisted higher degree $L$-function in the $q$-aspect. Specifically, let $L(s, F)$ be a standard $L$-function of degree $d\geq 3$. Under the above hypotheses, the bound \[ \sideset{}{^*}{\sum }_{{\chi \,({\rm mod}\, q)}}|L({1}/{2}, F\times \chi)|^2\ll_{F,\eta } \frac{q^{{d}/{2}}}{\log^{\eta }q} \] holds for some small $\eta \gt 0$.
