811 research outputs found
GROWTH OF PETERSSON INNER PRODUCTS OF FOURIER-JACOBI COEFFICIENTS OF SIEGEL CUSP FORMS (Analytic, geometric and -adic aspects of automorphic forms and -functions)
ON VALUES OF LOGARITHMIC DERIVATIVES OF -FUNCTIONS (Problems and Prospects in Analytic Number Theory)
This article is an extended version of a talk delivered at RJMS conference on "Problems and Prospects in Analytic Number Theory" held in November, 2020. In this note, we give a brief overview of the theme 'values of logarithmic derivatives of £-functions and zeta functions and its related topics'. We end by providing an outline of a recent work on values of logarithmic derivatives of £-functions attached to cuspidal elliptic Hecke eigenforms of integral weight
On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2
We investigate some key analytic properties of Fourier coefficients and Hecke
eigenvalues attached to scalar-valued Siegel cusp forms of degree 2, weight
and level . First, assuming that is a Hecke eigenform that is not of
Saito-Kurokawa type, we prove an improved bound in the -aspect for the
smallest prime at which its Hecke eigenvalue is negative. Secondly, we show
that there are infinitely many sign changes among the Hecke eigenvalues of
at primes lying in an arithmetic progression. Third, we show that there are
infinitely many positive as well as infinitely many negative Fourier
coefficients in any ``radial" sequence comprising of prime multiples of a fixed
fundamental matrix. Finally we consider the case when is of Saito--Kurokawa
type, and in this case we prove the (essentially sharp) bound for the
Fourier coefficients of whenever is squarefree,
confirming a conjecture made (in the case ) by Das and Kohnen.Comment: Minor corrections made; 38 page
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