On the Rankin--Selberg problem

Abstract

In this paper, we solve the Rankin--Selberg problem. That is, we break the well known Rankin--Selberg's bound on the error term of the second moment of Fourier coefficients of a $\mathrm{GL}(2)$ cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions.

28 pages. Comments are welcome!

References in corpus (1)

• Analytic twists of $\mathrm{G}{\mathrm{L}}_3\times \mathrm{G}{\mathrm{L}}_2$ automorphic forms

Cited by in corpus (10)

• Analytic twists of $\mathrm{G}{\mathrm{L}}_3\times \mathrm{G}{\mathrm{L}}_2$ automorphic forms
• Rankin-Selberg coefficients in large arithmetic progressions
• Second moment of degree three $L$-functions
• Hybrid subconvexity bounds for twists of $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions
• The cubic moment of Hecke--Maass cusp forms and moments of $L$-functions
• Uniform subconvex bounds for Rankin-Selberg $L$-functions
• Short Second Moment Bound for GL(2) $L$-functions in $q$-Aspect
• On the asymptotics of the shifted sums of Hecke eigenvalue squares
• Averages of coefficients of a class of degree 3 L-functions
• Shifted convolution sum with weighted average : $GL(3)\times GL(3)$ setup