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

TL;DR

This paper proves that a large proportion of points in certain vertical arithmetic progressions on the critical line are nonzeros of a fixed Dirichlet L-function, supporting the conjecture of linear independence of zeros' ordinates.

Contribution

It establishes a lower bound on the number of nonzero points in vertical arithmetic progressions and provides an upper bound for the first nonzero point, advancing understanding of zeros distribution.

Findings

At least a constant times T log T points are nonzeros in the progression.

Supports the conjecture that zeros' ordinates are linearly independent over rationals.

Provides bounds on the first nonzero point in the progression.

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 $L(s,\chi)$.