On a conjecture of Soundararajan

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.