The second moment of the $GL_3$ standard $L$-function on the critical line

TL;DR

This paper establishes a strong bound on the second moment of the $GL_3$ standard $L$-function on the critical line, leading to improvements in error terms and subconvexity bounds for related $L$-functions.

Contribution

It introduces a novel method building on recent work to bound the second moment of $GL_3$ $L$-functions, with applications to error terms and subconvexity bounds.

Findings

Improved error term in the Rankin-Selberg problem.

Enhanced subconvexity bounds for $GL_3 \times GL_2$ and $GL_3$ $L$-functions.

Estimate for averages of shifted convolution sums of $GL_3$ Fourier coefficients.

Abstract

We obtain a strong bound on the second moment of the $GL_3$ standard $L$-function on the critical line. The method builds on the recent work of Aggarwal, Leung, and Munshi which treated shorter intervals. We deduce some corollaries including an improvement on the error term in the Rankin-Selberg problem, and on certain subconvexity bounds for $GL_3 \times GL_2$ and $GL_3$ $L$-functions. As a byproduct of the method of proof, we also obtain an estimate for an average of shifted convolution sums of $GL_3$ Fourier coefficients.