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.
In a work in progress that I will talk about, Radziwill and I revisit the problem and generalize 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/hκ) exceptional x ∈ [X, 2X]).
We apply this result for instance to studying gaps between norm forms of an arbitrary number field.
This video is part of the Number Theory Web Seminar series.
