Random matrix theory and the Fourier coefficients of half-integral weight forms.
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Rankin-Selberg convolutions of generalized theta series.
Toshiaki Suzuki (1991)
Journal für die reine und angewandte Mathematik
Real zeros of holomorphic Hecke cusp forms
Amit Ghosh, Peter Sarnak (2012)
Journal of the European Mathematical Society
This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.
Real zeros of holomorphic Hecke cusp forms and sieving short intervals
Kaisa Matomäki (2016)
Journal of the European Mathematical Society
We study so-called real zeros of holomorphic Hecke cusp forms, that is, zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed the existence of many such zeros if many short intervals contain numbers whose prime factors all belong to a certain subset of the primes.We prove new results concerning this sieving problem which leads to improved lower bounds for the number of real zeros.
Recurrence Formulae for the Coefficients of Modular Forms and Congruences for the Partition Function and for the Coefficients of j(...), (j(...) - 1728)... and (j(...))... .
Torleiv Klove (1968)
Mathematica Scandinavica
Relative Kloosterman integrals for GL(3)
Hervé Jacquet, Yangbo Ye (1992)
Bulletin de la Société Mathématique de France
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