Exponential sums with coefficients of certain Dirichlet series

Abstract

Under the generalized Lindelöf Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski–Shapiro primes

Exponential sums with Dirichlet coefficients of L-functions

“…In [1], we proved Theorem 5 for c in the smaller range 1 < c < 25/24 under the Lindelöf Hypothesis (3.5).…”

“…which we then use to improve our recent result in [1] on squares of Hecke eigenvalues at Piatetski-Shapiro primes. In [1], we employed Vaughan's identity to relate exponential sums over primes to exponential sums over all integers in intervals.…”

“…which we then use to improve our recent result in [1] on squares of Hecke eigenvalues at Piatetski-Shapiro primes. In [1], we employed Vaughan's identity to relate exponential sums over primes to exponential sums over all integers in intervals. In this paper, we shall use a more direct approach to handle the prime condition, relating the said sums with logarithms of twists of F (s) with Dirichlet characters.…”

“…Our goal is to establish non-trivial bounds for exponential sums of the form n∼N c n e(f (n)), where c n = a n in the case of Theorem 1, and c n = a n if n is prime and c n = 0 otherwise in the case of Theorem 2. To this end, we split this exponential sum into short subsums in the same way as in [1]. To keep this paper self-contained, we copy this treatment from [1].…”

“…In this paper, by making several refinements, we improve our conditional estimate in [1] for exponential sums of the form n∼N a n e(f (n)), (1.1) where a n is the n-th coefficient of a Dirichlet series F (s) in a certain extension of the Selberg class (see section 2). In particular, we obtain a conditional improvement of a result of Jutila on exponential sums with Hecke eigenvalues.…”