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Revista Matemática Iberoamericana

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Volume 23, Issue 1, 2007, pp. 281–326
DOI: 10.4171/RMI/496

Published online: 2007-04-30

Small gaps in coefficients of $L$-functions and $\mathfrak{B}$-free numbers in short intervals

Emmanuel Kowalski[1], Olivier Robert[2] and Jie Wu[3]

(1) Université de Bordeaux I, Talence, France
(2) Université Jean-Monnet, Saint-Etienne, France
(3) Université Henri Poincaré, Vandoeuvre lès Nancy, France

We discuss questions related to the non-existence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog and Ono, and Alkan using new results about exponential sums and the distribution of $\mathfrak{B}$-free numbers.

Keywords: $\mathfrak{B}$-free numbers, Fourier coefficients of modular forms, Rankin-Selberg convolution, exponential sums

Kowalski Emmanuel, Robert Olivier, Wu Jie: Small gaps in coefficients of $L$-functions and $\mathfrak{B}$-free numbers in short intervals. Rev. Mat. Iberoam. 23 (2007), 281-326. doi: 10.4171/RMI/496