Exponential sums over primes are unbounded
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915501239697408 |
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| author | Bazin, Pierre-Alexandre |
| author_facet | Bazin, Pierre-Alexandre |
| contents | 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 Ramaré by proving the lower bound $$\sup_{n\le x} \left|\sum_{m\le n} Λ(m) \mathrm{e}(αm)\right| \gg x^{1/6 - \varepsilon}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18394 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential sums over primes are unbounded Bazin, Pierre-Alexandre Number Theory 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 Ramaré by proving the lower bound $$\sup_{n\le x} \left|\sum_{m\le n} Λ(m) \mathrm{e}(αm)\right| \gg x^{1/6 - \varepsilon}.$$ |
| title | Exponential sums over primes are unbounded |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.18394 |