On an open problem of Sankaranarayanan

Abstract

Let λ(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f (z) ∈ S k (Γ) for the full modular group. In one of his papers, Sankaranarayanan mentioned that it is an open problem to give a non-trivial estimate for the sum of λ(n) over cubes, i.e. n x λ(n 3 ). In this paper, we are able to use the analytic properties of symmetric power L-functions to solve his problem. More precisely, we prove that n x λ(n 3 ) x 3 4 +ε , n x λ(n 4 ) x 7 9 +ε .

Ω-Result on Coefficients of Automorphic L-Functions Over Sparse Sequences

“…On the other hand, the sum over squares n≤x λ f (n 2 ) was considered in Ivić [5], Fomenko [2] and Sankaranarayanan [20]. Lü [14] obtained the bound of n≤x λ f (n j ) (j = 3, 4). Lao and Sankaranarayanan [10] established the asymptotic formula of the sum n≤x λ 2 f (n j ), where j = 2, 3, 4.…”

A note on signs of Fourier coefficients of two cusp forms

“…Proof. We refer [5] and [14] for the proof. Now comparing 1 − β j and α j in 4.1 and 4.2, we can see that the bounds contradict each other.…”

A short note on sign changes

“…In fact, we give a lower bound for the number of sign changes in the interval (x, 2x]. We use the results of O. M. Fomenko [3], H. Lao and A. Sankaranarayanan [10] and G. S. Lü [11] to prove this result.…”

“…Proof of Theorem 2.1 Let us denote 1/2, 3/4 and 7/9 by β j for j = 2, 3 and 4 respectively. For any ǫ > 0, [3] and [11] give the following estimates.…”