Discrete mean square of the coefficients of triple product L-functions over certain sparse sequence

Abstract

Let f,g and h be three distinct normalized primitive holomorphic cusp forms of even integral weights k1,k2 and k3 for the full modular group SL(2,Z), respectively. And let λf×g×h(n) denote the nth coefficient of triple product L-function L(f×g×h,s) associated to f,g,h. In this paper, we are interested in the average behavior of the following sum∑a2+b2+c2+d2≤x(a,b,c,d)∈Z4λf×g×h2(a2+b2+c2+d2) for x≥x0, where x0 is some sufficiently large number.