Speaker: Dr. LIN Yongxiao (Shandong University)
Venue: Online
Tencent Meeting ID: 321-4296-7396
Abstract: Let $(\lambda_f(n))_{n\geq 1}$ be the Fourier coefficients of a Hecke--Maass $f$ on $GL_2$. The classical Rankin--Selberg problem asks for a better error term in the asymptotic formula for $\sum_{n\leq X} \lambda_f(n)^2$, as $X\to \infty$. We study an arithmetic analogue of this problem where the coefficients are restricted to arithmetic progressions $n\equiv a (mod q)$ with varying moduli $q$ (as $q\to \infty$). The functional equation for the related L-functions (subject to the Ramanujan--Petersson conjecture) implies a level of distribution 2/5. We explain how one can obtain a better exponent $2/5+\eta$ (some $\eta>0$). Key to our proof is convolution identity for the coefficients of the L-function $L(f\times f,s)$. This is a joint work with E. Kowalski and Ph. Michel.