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On higher moments of Hecke eigenvalues attached to cusp forms

Guodong Hua (2022)

Czechoslovak Mathematical Journal

Let f , g and h be three distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 and k 3 for the full modular group Γ = SL ( 2 , ) , respectively, and let λ f ( n ) , λ g ( n ) and λ h ( n ) denote the n th normalized Fourier coefficients of f , g and h , respectively. We consider the cancellations of sums related to arithmetic functions λ g ( n ) , λ h ( n ) twisted by λ f ( n ) and establish the following results: n x λ f ( n ) λ g ( n ) i λ h ( n ) j f , g , h , ε x 1 - 1 / 2 i + j + ε for any ε > 0 , where 1 i 2 , j 5 are any fixed positive integers.

On Hilbert modular forms modulo p: explicit ring structure.

Shoyu Nagaoka (2006)

Revista Matemática Iberoamericana

H. P. F. Swinnerton-Dyer determined the structure of the ring of modular forms modulo p in the elliptic modular case. In this paper, the structure of the ring of Hilbert modular forms modulo p is studied. In the case where the discriminant of corresponding quadratic field is 8 (or 5), the explicit structure is determined.

On Lehmer's problem and Dedekind sums

Xiaowei Pan, Wenpeng Zhang (2011)

Czechoslovak Mathematical Journal

Let p be an odd prime and c a fixed integer with ( c , p ) = 1 . For each integer a with 1 a p - 1 , it is clear that there exists one and only one b with 0 b p - 1 such that a b c (mod p ). Let N ( c , p ) denote the number of all solutions of the congruence equation a b c (mod p ) for 1 a , b p - 1 in which a and b ¯ are of opposite parity, where b ¯ is defined by the congruence equation b b ¯ 1 ( mod p ) . The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet L -functions to study the hybrid mean value problem involving...

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