Geodesic
Another application of Linnik dispersion method
Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 148-163.

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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. 
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Étienne Fouvry; Maksym Radziwiłł. Another application of Linnik dispersion method. Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 148-163. http://geodesic.mathdoc.fr/item/CHEB_2018_19_3_a12/

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