On a conjecture of Soundararajan

TL;DR

This paper proves a conjecture by Soundararajan on the distribution of smooth numbers in residue classes for certain moduli, leveraging recent advances in zero-free regions of $L$-functions and properties of characters with Siegel zeros.

Contribution

It establishes Soundararajan's conjecture for smooth numbers over reduced residue classes for moduli with specific zero-free region properties, including those with Siegel zeros.

Findings

Proves the conjecture for moduli with at most one problematic character.

Shows the conjecture holds for moduli with Siegel zeros.

Utilizes zero-free regions and the Deuring-Heilbronn phenomenon in the proof.

Abstract

Building on recent work of A. Harper (2012), and using various results of M. C. Chang (2014) and H. Iwaniec (1974) on the zero-free regions of $L$-functions $L(s,\chi)$ for characters $\chi$ with a smooth modulus $q$, we establish a conjecture of K. Soundararajan (2008) on the distribution of smooth numbers over reduced residue classes for such moduli $q$. A crucial ingredient in our argument is that, for such $q$, there is at most one "problem character" for which $L(s,\chi)$ has a smaller zero-free region. Similarly, using the "Deuring-Heilbronn" phenomenon on the repelling nature of zeros of $L$-functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros.