# Multiplicative functions in short intervals

Date: Thu, Oct 6, 2022

Location: PIMS, University of British Columbia, PIMS, University of Lethbridge, PIMS, University of Northern British Columbia, Zoom, Online

Conference: L-Functions in Analytic Number Theory Seminar

Subject: Mathematics, Number Theory

Class: Scientific

### Abstract:

In this talk, we are interested in a general class of multiplicative functions. For a function that belongs to this class, we will relate its “short average” to its “long average”. More precisely, we will compute the variance of such a function over short intervals by using Fourier analysis and by counting rational points on certain binary forms. The discussion is applicable to some interesting multiplicative functions such as

$$

\mu_k(n), \frac{\phi (n)}{n}, \frac{n}{\phi (n)}, \mu^2(n)\frac{\phi(n)}{n},

\sigma_\alpha (n), (-1)^{\#\left\{p: p^k | n \right\}}

$$

and many others and it provides various new results and improvements to the previous result

in the literature. This is a joint work with Mithun Kumar Das.

This event is part of the PIMS CRG Group on *L*-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar