On the second integral moment of $L$-functions

TL;DR

Assuming the generalized Ramanujan conjecture, the paper derives a refined bound on the second moment of automorphic L-functions on GL(d) over Q, improving understanding of their size on the critical line.

Contribution

The paper provides a small log-saving bound on the second integral moment of automorphic L-functions assuming the Ramanujan conjecture for GL(d), with d ≥ 3.

Findings

Bound  T^{d/2} / ( ext{log} T)^{\u03b7_d} for the second moment

Improves understanding of the size of L-functions on the critical line

Assumes the generalized Ramanujan conjecture for automorphic forms

Abstract

Assume that the generalized Ramanujan conjecture holds on the automorphic $L$-function $L(s, \pi)$ on $\GL_d$ over $\mathbb{Q}$ with $d\geq 3$, we can obtain a small log-saving non-trivial bound on the second integral moment of $L(1/2+it, \pi)$. Specifically the bound \[ \int_{T}^{2T}\Big|L\big(\frac{1}{2}+it, \pi\big)\Big |^2 \dd t\ll_{\pi} \frac{T^{\frac{d}{2}}}{\log^{\eta_d}T} \] holds for a small constant $\eta_d>0$.