On Hecke eigenvalues at Piatetski-Shapiro primes

Abstract

Abstract. Let λ(n) be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant C > 0 depending on the cusp form and every fixed c in the range 1 < c < 8/7, the mean value of λ(p) is ≪ exp(−C √ log N ) as p runs over all (Piatetski-Shapiro) primes of the form [n c ] with n ∈ AE and n ≤ N .

Exponential sums with coefficients of certain Dirichlet series

“…To this end, we shall split this exponential sum into short subsums using a Farey dissection of a certain interval. We note that the splitting of the summation interval in the present paper differs from that in [1]. It will become clear in the next section why it is advantageous to split the summation interval as described below.…”

“…According to [1], Theorem 3 and (2.5) imply the following result on the sign changes of λ(p) at Piatetski-Shapiro primes p.…”

“…Hence, we have The ordinary Piatetski-Shapiro prime number theorem (see [1], for example) tells us that…”

“…to be chosen later. Now we make a Farey dissection of level Q of the interval [h(N ′ ), h(N )) (for details on Farey intervals, see [1], for example). In this way, we write [h(N ′ ), h(N )) as the disjoint union of intervals of the form…”

On Hecke eigenvalues at primes of the form [g(n)]

“…The estimation of mean-values of arithmetic functions over sparse sequences and the detection of primes in arithmetically interesting and sparse sets of natural numbers are often very hard and of great interest to analytic number theorists. In [1], we investigated a problem that addresses both of these questions, namely the distribution of Fourier coefficients of cusp forms for the full modular group at Piatetski-Shapiro primes. These are primes of the form [n c ], where c > 1 is fixed.…”