Bounds for exponential sums with random multiplicative coefficients
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908634443677696 |
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| author | Hardy, Seth |
| author_facet | Hardy, Seth |
| contents | For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{θ\in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n θ) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N \rightarrow \infty$. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\varepsilon > 0$, we prove the partial upper bound result $$ \max_{θ\in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{\substack{n \leq N \\ P(n) \geq N^{0.8}}} f(n) \mathrm{e} (n θ) \Bigr| \ll {(\log N)}^{7/4 + \varepsilon},$$ asymptotically almost surely as $N \rightarrow \infty$, where $P(n)$ denotes the largest prime factor of $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16256 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds for exponential sums with random multiplicative coefficients Hardy, Seth Number Theory 11K65, 42A05 For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{θ\in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n θ) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N \rightarrow \infty$. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\varepsilon > 0$, we prove the partial upper bound result $$ \max_{θ\in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{\substack{n \leq N \\ P(n) \geq N^{0.8}}} f(n) \mathrm{e} (n θ) \Bigr| \ll {(\log N)}^{7/4 + \varepsilon},$$ asymptotically almost surely as $N \rightarrow \infty$, where $P(n)$ denotes the largest prime factor of $n$. |
| title | Bounds for exponential sums with random multiplicative coefficients |
| topic | Number Theory 11K65, 42A05 |
| url | https://arxiv.org/abs/2401.16256 |