Multiplicative functions in short intervals revisited

Abstract: A few years ago Maksym Radziwill and I showed that the average of a multiplicative function in almost all very short intervals $[x, x+h]$ is close to its average on a long interval $[x, 2x]$. This result has since been utilized in many applications. I will talk about recent work, where Radziwill and I revisit the problem and generalise our result to functions which vanish often as well as prove a power-saving upper bound for the number of exceptional intervals (i.e. we show that there are $O(X/h^\kappa)$ exceptional $x \in [X, 2X]$). We apply this result for instance to studying gaps between norm forms of an arbitrary number field.

number theory

Audience: researchers in the topic

Harvard number theory seminar