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arXiv · 1812.00562

Another application of Linnik's dispersion method

Étienne Fouvry·Maksym Radziwiłł
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Abstract

Let $α_m$ and $β_n$ be two sequences of real numbers supported on $[M, 2M]$ and $[N, 2N]$ with $M = X^{1/2 - δ}$ and $N = X^{1/2 + δ}$. We show that there exists a $δ_0 > 0$ such that the multiplicative convolution of $α_m$ and $β_n$ has exponent of distribution $\frac{1}{2} + δ-\varepsilon$ (in a weak sense) as long as $0 \leq δ< δ_0$, the sequence $β_n$ is Siegel-Walfisz and both sequences $α_m$ and $β_n$ 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.

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Étienne Fouvry, Maksym Radziwiłł. 2018-12-04. Another application of Linnik's dispersion method. https://arxiv.org/abs/1812.00562

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