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

Xue Han, Xiaofei Yan and Deyu Zhang
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Xue Han: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
Xiaofei Yan: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
Deyu Zhang: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China

Mathematics, 2021, vol. 9, issue 11, 1-13

Abstract: Let P c ( x ) = { p ? x | p , [ p c ] are primes } , c ? R + ? N and ? s y m 2 f ( n ) be the n -th Fourier coefficient associated with the symmetric square L -function L ( s , s y m 2 f ) . For any A > 0 , we prove that the mean value of ? s y m 2 f ( n ) over P c ( x ) is ? x log ? A ? 2 x for almost all c ? ? , ( 5 + 3 ) / 8 ? ? 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 ? f 2 ( n ) over P c ( x ) is ? p , q p r i m e p ? x , q = [ p c ] ? f 2 ( p ) = x c log 2 x ( 1 + o ( 1 ) ) , for almost all c ? ? , ( 5 + 3 ) / 8 ? ? , where ? f ( n ) is the normalized n -th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

Keywords: primes; Fourier coefficient; symmetric square L -function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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