Kaisa Matomäki: Multiplicative functions in short intervals revisited (NTWS 009)
Channel: Number Theory Web Seminar
Date: 11/04/2023

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.
In a work in progress that I will talk about, 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 Web Seminar: https://www.ntwebseminar.org

Original air date: 
Tuesday, June 2, 2020 (1am PDT, 4am EDT, 9am BST, 10am CEST, 1:30pm IST, 4pm China Standard Time, 6pm AEST, 8pm NZST)