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On sums of Hecke series in short intervals

Aleksandar Ivić (2001)

Journal de théorie des nombres de Bordeaux

We have K - G k j K + G α j H j 3 ( 1 2 ) ϵ G K 1 + ϵ for K ϵ G K , where α j = ρ j ( 1 ) 2 ( cosh π k j ) - 1 , and ρ j ( 1 ) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue λ j = k j 2 + 1 4 to which the Hecke series H j ( s ) is attached. This result yields the new bound H j ( 1 2 ϵ k j 1 3 + ϵ .

On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua (2022)

Czechoslovak Mathematical Journal

Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = SL ( 2 , ) . Denote by λ f ( n ) the n th normalized Fourier coefficient of f . We are interested in the average behaviour of the sum a 2 + b 2 x λ f j ( a 2 + b 2 ) for x 1 , where a , b and j 9 is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power L -functions and Rankin-Selberg L -functions.

Oscillations of Hecke eigenvalues at shifted primes.

Liangyi Zhao (2006)

Revista Matemática Iberoamericana

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is √n at prime arguments.

Produits de Petersson de formes modulaires associées aux valeurs de fonctions L

Lionel Fourquaux (2002)

Journal de théorie des nombres de Bordeaux

Considérons les formes linéaires f L ( f , χ , 1 ) sur l’espace vectoriel des formes paraboliques de poids 2 pour le groupe de congruence Γ 0 ( p ) , avec χ un caractère de Dirichlet modulo p . Par le produit scalaire de Petersson, on peut leur associer des formes paraboliques. Cet article détermine le produit scalaire de deux de ces formes, pour deux caractères de Dirichlet non triviaux de parités différentes.

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