On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Xue Han,Xiaofei Yan,Deyu Zhang

doi:10.3390/math9111254

Abstract

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

Highlights

Let k be an even positive integer, f be a holomorphic cusp form of weight k for the full modular group and λ f (n) be the normalized n-th Fourier coefficient of f, i.e., Piatetski-Shapiro Prime Twins. ∞ f (z) =Mathematics 2021, 9, 1254. https://Received: 28 April 2021Accepted: 26 May 2021Published: 30 May 2021Publisher’s Note: MDPI stays neutral e(nz).If we assume that f is an eigenform of all the Hecke operators, f can be normalized such that λ f (1) = 1 and λ f (n) is real
By using the analytic properties of the Rankin–Selberg L-function L(s, f ⊗ f). Another interesting question considered by many authors is the mean value of λ f (n) over certain sets of primes
Baier and Zhao [24] studied the distribution of λ f (n) at Piatetski–Shapiro primes by considering estimates of exponential sum involving

Summary

Introduction

Let k be an even positive integer, f be a holomorphic cusp form of weight k for the full modular group and λ f (n) be the normalized n-th Fourier coefficient of f , i.e., Piatetski-Shapiro Prime Twins. As N → ∞, log N by using the analytic properties of the Rankin–Selberg L-function L(s, f ⊗ f) Another interesting question considered by many authors is the mean value of λ f (n) over certain sets of primes. Baier and Zhao [24] studied the distribution of λ f (n) at Piatetski–Shapiro primes by considering estimates of exponential sum involving. Assuming the Riemann Hypothesis of automorphic L-function L(s, f ) is true, they found that (2) holds for all c ∈ (0, 1). We consider the mean value of Fourier coefficients of symmetric square Lfunction over Piatetski–Shapiro prime twins and obtain the following results, which imply a result on the distribution of λ2f (n) at Piatetski–Shapiro prime twins.

Auxiliary Lemmas

Proof of Theorem 2

Proof of Theorem 1