Discorrelation of the Möbius function with linear phases in short intervals

Source 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$H$.