Remarks on Landau-Siegel zeros

TL;DR

This paper shows that for specific families of $L$-functions, having only real zeros near 1 allows for significant improvements in understanding Landau-Siegel zeros, extending previous restrictive results.

Contribution

It establishes new results linking the distribution of zeros of $L$-functions near 1 to the properties of Landau-Siegel zeros, generalizing prior restrictive conditions.

Findings

Improved bounds on Landau-Siegel zeros under certain zero distribution assumptions

Extension of previous results by Sarnak and others to broader conditions

Enhanced understanding of the zero-free regions for $L$-functions

Abstract

For certain families of $L$-functions, we prove that if each $L$-function in the family has only real zeros in a fixed yet arbitrarily small neighborhood of $s=1$, then one may considerably improve upon the known results on Landau-Siegel zeros. Sarnak and the third author proved a similar result under much more restrictive hypotheses.