Nonzero values of Dirichlet $L$-functions in vertical arithmetic progressions

Abstract

Let $L(s,\chi )$ be a fixed Dirichlet $L$-function. Given a vertical arithmetic progression of $T$ points on the line $\Re (s)=1/2$, we show that $\gg T\log T$ of them are not zeros of $L(s,\chi )$. This result provides some theoretical evidence towards the conjecture that all ordinates of zeros of Dirichlet $L$-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of $L(s,\chi )$.

25 pages. To appear in Int. J. Number Theory. New version has some minor changes, including adding a mention of the recent paper of Li and Radziwiłł (arXiv:1208.2684)