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A note on average behaviour of the Fourier coefficients of j th symmetric power L -function over certain sparse sequence of positive integers

Youjun Wang — 2024

Czechoslovak Mathematical Journal

Let j 2 be a given integer. Let H k * be the set of all normalized primitive holomorphic cusp forms of even integral weight k 2 for the full modulo group SL ( 2 , ) . For f H k * , denote by λ sym j f ( n ) the n th normalized Fourier coefficient of j th symmetric power L -function ( L ( s , sym j f ) ) attached to f . We are interested in the average behaviour of the sum n = a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 x ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) 6 λ sym j f 2 ( n ) , where x is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).

Mean values related to the Dedekind zeta-function

Hengcai TangYoujun Wang — 2024

Czechoslovak Mathematical Journal

Let K / be a nonnormal cubic extension which is given by an irreducible polynomial g ( x ) = x 3 + a x 2 + b x + c . Denote by ζ K ( s ) the Dedekind zeta-function of the field K and a K ( n ) the number of integral ideals in K with norm n . In this note, by the higher integral mean values and subconvexity bound of automorphic L -functions, the second and third moment of a K ( n ) is considered, i.e., n x a K 2 ( n ) = x P 1 ( log x ) + O ( x 5 / 7 + ϵ ) , n x a K 3 ( n ) = x P 4 ( log x ) + O ( X 321 / 356 + ϵ ) , where P 1 ( t ) , P 4 ( t ) are polynomials of degree 1, 4, respectively, ϵ > 0 is an arbitrarily small number.

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