Discrete mean square of the coefficients of symmetric square L-functions on certain sequence of positive numbers

Abstract

In this paper, we will be concerned with the average behavior of the nth normalized Fourier coefficients of symmetric square L-function (i.e., L(s, sym2f)) over certain sequence of positive integers. Precisely, we prove an asymptotic formula for ∑a2+b2+c2+d2≤x(a,b,c,d)∈Z4λsym2f2(a2+b2+c2+d2),where x≥ x (sufficiently large), and L(s,sym2f):=∑n=1∞λsym2f(n)ns.