Real zeros of real odd Dirichlet $L$-functions

Abstract:

Let $\chi$ be a real odd Dirichlet character of modulus $d$, and let $L(s,\chi )$ be the associated Dirichlet $L$-function. As a consequence of the work of Low and Purdy, it is known that if $d\le 800 000$ and $d\neq 115 147$, $357 819$, $636 184$, then $L(s,\chi )$ has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if $d\le 300 000 000$, then $L(s,\chi )$ has no positive real zeros.