Discorrelation of the Möbius function with linear phases in short intervals

Javier Pliego; Mengdi Wang

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arXiv · 2610.11487

Discorrelation of the Möbius function with linear phases in short intervals

Javier Pliego·Mengdi Wang

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Abstract

We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequency over short intervals. As an application, we prove that \[ \Big|\sum_{x 0$ unless there exists an integer $1\leq q\ll (\log x)^{O_A(1)}$ such that $\|qα\| \ll x (\log x)^{O_A(1)}/H^2$. This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials would lead to a corresponding improvement in the lower bound for $H$.

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Javier Pliego, Mengdi Wang. 2026-10-08. Discorrelation of the Möbius function with linear phases in short intervals. https://arxiv.org/abs/2610.11487

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