General asymptotic formula of Fourier coefficients of cusp forms over sum of two squares

Abstract

Let λf(n) denote the nth normalized Fourier coefficient of the primitive holomorphic cusp forms of even integral weight k≥2 for the full modular group SL(2,Z). For x≥1, we are interested in the sums ∑n≤xλf(n)ℓ and ∑a2+b2≤xλf(a2+b2)ℓ. In this paper, we are able to establish the asymptotic formulae for general cases of the power sum for every positive integer ℓ∈N. The special cases of our results improve previous results.