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

Let P c ( x ) = { p ≤ x | p , [ p c ] are primes } , c ∈ R + ∖ N and λ 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 λ s y m 2 f ( n ) over P c ( x ) is ≪ x log − A − 2 x 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 λ f 2 ( n ) over P c ( x ) is ∑ p , q p r i m e p ≤ x , q = [ p c ] λ f 2 ( p ) = x c log 2 x ( 1 + o ( 1 ) ) , for almost all c ∈ ε , ( 5 + 3 ) / 8 − ε , where λ f ( n ) is the normalized n -th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.