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On a conjecture of Soundararajan

On a conjecture of Soundararajan Building on recent work of Harper, and using various results of Chang and Iwaniec on the zero‐free regions of L$L$‐functions L(s,χ)$L(s,\chi )$ for characters χ$\chi$ with a smooth modulus q$q$, we establish a conjecture of Soundararajan on the distribution of smooth numbers over reduced residue classes for such moduli q$q$. A crucial ingredient in our argument is that, for such q$q$, there is at most one ‘problem character’ for which L(s,χ)$L(s,\chi )$ has a smaller zero‐free region. Similarly, using the ‘Deuring–Heilbronn’ phenomenon on the repelling nature of zeros of L$L$‐functions close to one, we also show that Soundararajan's conjecture holds for a family of moduli having Siegel zeros. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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Publisher
Wiley
Copyright
© 2022 London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms.12524
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Feb 1, 2022

Keywords: 11N25; 11N69 (primary); 11M20 (secondary)

References