Year: 2025
Type: preprint
Abstract: We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} Λ(m) \mathrm{e}(αm)\right|^2 \gg y\log y$$ whenever $y \ll x^{1/3-\varepsilon}.$ This answers a question of Aymone and Ramaré by proving the lower bound $$\sup_{n\le x} \left|\sum_{m\le n} Λ(m) \mathrm{e}(αm)\right| \gg x^{1/6 - \varepsilon}.$$
Source: arXiv (Cornell University)
Author: Bazin, Péa
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7077860424
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