Random multiplicative functions and typical size of character in short intervals
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910324282621952 |
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| author | Caich, Rachid |
| author_facet | Caich, Rachid |
| contents | We examine the conditions under which the sum of random multiplicative functions in short intervals, given by $\sum_{x<n \leqslant x+y} f(n)$, exhibits the phenomenon of \textit{better than square-root cancellation}. We establish that the point at which the square-root cancellation diminishes significantly is approximately when the ratio $\log\big(\frac{x}{y}\big)$ is around $\sqrt{\log\log x}$. By modeling characters by random multiplicative functions, we give a sharp bound of $\frac{1}{r-1}\sum_{χ\!\!\!\mod r} \big|\sum_{x<n\leqslant x+y}χ(n)\big|$, where $r$ is a large prime and $x+y\leqslant r $. This extends the result of Harper \cite{Harper_charac}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06426 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random multiplicative functions and typical size of character in short intervals Caich, Rachid Number Theory We examine the conditions under which the sum of random multiplicative functions in short intervals, given by $\sum_{x<n \leqslant x+y} f(n)$, exhibits the phenomenon of \textit{better than square-root cancellation}. We establish that the point at which the square-root cancellation diminishes significantly is approximately when the ratio $\log\big(\frac{x}{y}\big)$ is around $\sqrt{\log\log x}$. By modeling characters by random multiplicative functions, we give a sharp bound of $\frac{1}{r-1}\sum_{χ\!\!\!\mod r} \big|\sum_{x<n\leqslant x+y}χ(n)\big|$, where $r$ is a large prime and $x+y\leqslant r $. This extends the result of Harper \cite{Harper_charac}. |
| title | Random multiplicative functions and typical size of character in short intervals |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.06426 |