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arXiv · 2402.06426

Random multiplicative functions and typical size of character in short intervals

Rachid Caich
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Abstract

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 $\frac{1}{r-1}\sum_{\chi \!\!\!\mod r} \big|\sum_{x<n\leqslant x+y}\chi(n)\big|$, where $r$ is a large prime and $x+y\leqslant r $. This extends the result of Harper \cite{Harper_charac}.

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Rachid Caich. 2024-02-09. Random multiplicative functions and typical size of character in short intervals. https://arxiv.org/abs/2402.06426

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