On the second integral moment of $L$-functions

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(\frac{1}{2}+it, \pi)\Big|^2 d t\ll_{\pi} \frac{T^{\frac{d}{2}}}{\log^{\eta_d}T} ] holds for a small constant $\eta_d>0$. As an application, we give a new asymptotic formula for the average of the coefficient $\lambda_{1\boxplus \pi}(n)$.