The behavior of Riesz means of the coefficients of a~symmetric square $L$-function

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 274-286

Résumé

Let $f(z)$ be a holomorphic Hecke eigencuspform of even weight $k$ with respect to $SL(2,\mathbb Z)$ and let $L(s,\mathrm{sym}^2f)=\sum_{n=1}^\infty c_nn^{-s}$, $\operatorname{Re}s>1$, be the symmetric square $L$-function associated with $f$. Represent the Riesz mean $(\rho\ge 0)$ $$ \frac1{\Gamma(\rho+1)}\sum_{n\le x}'(x-n)^\rho c_n=:D_{\rho}(x;\mathrm{sym}^2 f) $$ as the sum of the “residue function” $\Gamma(\rho+1)^{-1}L(0,\mathrm{sym}^2f)x^\rho$ and the “error term” $$ D_\rho(x;\mathrm{sym}^2f)=\frac{L(0,\mathrm{sym}^2f)}{\Gamma(\rho+1)}x^\rho+\Delta_\rho(x;\mathrm{sym}^2f). $$ Using the Voronoi formula for $\Delta_\rho(x;\mathrm{sym}^2f)$, obtained earlier (see Zap. Nauchn. Semin. POMI. 314, 247–256 (2004)), the integral $$ \int_1^X\Delta_\rho{(x;\mathrm{sym}^2f)}^2\,dx, $$ is estimated. In this way, an asymptotics for $0\rho\leqslant1$ and an upper bound for $\rho=0$ are obtained. Also the existence of a limiting distribution for the function $$ x^{-\frac23\rho-\frac13}\Delta_\rho(x;\mathrm{sym}^2f), \quad 0\rho\leqslant1, $$ and, as a corollary, for the function $$ x^{-\frac23\rho-\frac13}D_{\rho}(x;\mathrm{sym}^2f), \quad 0\rho1. $$ is established. Bibliography: 12 titles.