This article is cited in 1 paper

Another application of Linnik dispersion method

Étienne Fouvry^abc, Maksym Radziwiłł<sup>d

a</sup> Laboratoire de Mathématiques d'Orsay, Univ. Paris–Sud
^b Université Paris–Saclay, 91405 Orsay, France
^c CNRS
^d Department of Mathematics, McGill University, Burnside Hall, Room 1005, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 0B9

Abstract: Let $\alpha_m$ and $\beta_n$ be two sequences of real numbers supported on $[M, 2M]$ and $[N, 2N]$ with $M = X^{1/2 - \delta}$ and $N = X^{1/2 + \delta}$. We show that there exists a $\delta_0 > 0$ such that the multiplicative convolution of $\alpha_m$ and $\beta_n$ has exponent of distribution $\frac{1}{2} + \delta-\varepsilon$ (in a weak sense) as long as $0 \leq \delta < \delta_0$, the sequence $\beta_n$ is Siegel-Walfisz and both sequences $\alpha_m$ and $\beta_n$ are bounded above by divisor functions. Our result is thus a general dispersion estimate for “narrow” type-II sums. The proof relies crucially on Linnik's dispersion method and recent bounds for trilinear forms in Kloosterman fractions due to Bettin-Chandee. We highlight an application related to the Titchmarsh divisor problem.

Keywords: equidistribution in arithmetic progressions, dispersion method.

UDC: 512.54

Received: 22.06.2018
Accepted: 10.10.2018

Language: English

DOI: 10.22405/2226-8383-2018-19-3-148-163

Bibliographic databases:

•