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

Greg Martin; Nathan Ng

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

Number Theory 2012-08-17 v2

Authors: Greg Martin , Nathan Ng

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 $L(s,\chi)$ .

Keywords

dirichlet characters l-functions arithmetic progressions

Cite

@article{arxiv.1109.1788,
  title  = {Nonzero values of Dirichlet $L$-functions in vertical arithmetic progressions},
  author = {Greg Martin and Nathan Ng},
  journal= {arXiv preprint arXiv:1109.1788},
  year   = {2012}
}

Comments

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 Radziwi{\l}{\l} (arXiv:1208.2684)

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

Abstract

Keywords

Cite

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