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 $\lambda_{ 1\boxplus \pi}(n)$.
Categories: math.NT