The distribution of Fourier coefficients of cusp forms over sparse sequences

Abstract

Let λf(n) be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈ Sk(Γ). We establish that Σn≤λf2(nj) = cjx + O (x1-2/(j+1)2+ 1) for = 2,3,4, which improves the previous results. For j = 2, we even establish a better result.