On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms

Abstract

Abstract. Let S k (Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group. Let λ f (n) and λg (n) be the n-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms f (z), g(z) ∈ S k (Γ), respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.

Integral power sums of Hecke eigenvalues

“…The main idea for our improvement is an alternative expression of G (s) in Lemma 2.1 below, different from [15,16,17,13]; this expression decomposes G (s) into a product of L-functions, general and (more importantly) of lower degree ( 3). Hence we can take advantage of their (individual or averaged) subconvexity bounds (see…”

Average estimates and sign change of Fourier coefficients of cusp forms at integers represented by binary quadratic form of fixed discriminant

“…O. M. Fomenko [5] has established estimates for n≤x (λ f (n)) r , when r = 2, 4 and improved the result of Moreno and Shahidi [16]. G. Lü [[13], [14], [15]] improved Fomenko's results and generalised the results for higher moments, i.e., for r ≥ 2. Y. K. Lau et al [11] considered more general cases and obtained better estimates.…”

Estimate for Higher Moments of Cusp Form Coefficients Over Sum of Two Squares Coefficients Over Sum of Two Squares

“…where c f is some suitable constant depending on f . Later, other authors considered the higher moments of the Fourier coefficients of cusp forms (see c.f., [18,7,22]). In particular, Lau and Lü [19] established a general formula for the summatory function…”