Random multiplicative functions and typical size of character in short intervals

Abstract

We examine the conditions under which the sum of random multiplicative functions in short intervals, given by ${\sum }_{x<n⩽x+y}f(n)$, 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 $\log \left(\frac{x}{y}\right)$ is around $\sqrt{\log \log x}$. By modeling characters by random multiplicative functions, we give a sharp bound of $\frac{1}{r-1}{\sum }_{\chi \mathrm{mod}r}|{\sum }_{x<n⩽x+y}\chi (n)|$, where $r$ is a large prime and $x+y⩽r$. This extends the result of Harper \cite{Harper_charac}.

Cited by in corpus (1)

• Harper's beyond square-root conjecture