Another application of Linnik's dispersion method

Abstract

Let be α_m and β_n---two sequences of real numbers supported on segments [M,2M] and [N,2N], where M = X^(1/2 − δ) and N = X^(1/2 + δ)... We prove the existence of such a constant δ₀ that the multiplicative convolution α_m and β_n has a distribution level 1/2 + δ - ε (in a weak sense), if only 0 ⩽ δ < δ₀, subsequence β_n is a Siegel-Walvis sequence, and both sequences α_m and β_n are bounded from above by the divisor function. Our result, therefore, is the overall variance estimate for "short", type II sums. The proof makes essential use of Linnik's dispersion method and recent estimates for trilinear sums with Kloosterman fractions due to Bettin and Chandy. We will also focus on the application of this result to the Titchmarsh divisor problem.

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