Second moment of degree three $L$-functions

Abstract

Let $F$ be a Hecke-Maaß cusp form for $\mathrm{SL}(3,\mathbb{Z})$. We obtain the first non-trivial upper bound of the second moment of $L(F,s)$ in $t$-aspect: $${\int }_T^{2T}|L(F,1/2+it){|}^{2}dt{\ll }_{F,\epsilon }{T}^{3/2-3/32+\epsilon }.$$ Immediate corollaries include improvements over the existing results on the subconvexity bound for self-dual $\mathrm{GL}(3)$ $L$-functions in the $t$-aspect and for self-dual $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions in the $\mathrm{GL}(2)$ spectral aspect, the error term in the Rankin-Selberg problem, and the zero density estimate for $\mathrm{GL}(3)$ $L$-functions.

arXiv commentFinal Draft

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• Distribution of mass of holomorphic cusp forms
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Cited by in corpus (1)

• Strong Hybrid Subconvexity for Twisted Selfdual ${\mathrm{GL}}_3$ $L$-Functions