Mathematics Colloquium
Speaker: Akshaa Vatwani, IIT Gandhinagar
Host: Keshav Aggarwal
Title: Multiplicative functions in arithmetic progressions
Time, day and date: 4:00:00 PM – 5:00:00 PM, Wednesday, December 10
Venue: Ramanujan Hall
Abstract: Given an arithmetic function $f:\mathbb N \to \mathbb C$, it is of interest to study whether it is uniformly distributed in residue classes. A well-known example is the Bombieri Vinogradov theorem, which establishes that the primes are equidistributed in arithmetic progressions on average with a level of distribution $1/2$.
In this talk, we discuss a mean-value estimate for trilinear forms involving arbitrary sequences over arithmetic progressions, excluding the contribution of exceptional characters. We prove upper bounds in terms of the $L^{2}$-norms of the corresponding sequences.
As an application, we obtain a Bombieri-Vinogradov type theorem for a broad class of multiplicative functions supported on smooth numbers. In particular, we show that these functions are equidistributed in arithmetic progressions on average over moduli $q \le x^{3/5-\varepsilon}$, if they satisfy a Siegel-Walfisz criterion. This is joint work with Aditi Savalia and Arindam Roy.