Self-approximation of Dirichlet L-functions

Abstract

Let $d$ be a real number, let $s$ be in a fixed compact set of the strip $1/2<\sigma <1$, and let $L(s,\chi )$ be the Dirichlet $L$-function. The hypothesis is that for any real number $d$ there exist 'many' real numbers $\tau$ such that the shifts $L(s+i\tau ,\chi )$ and $L(s+id\tau ,\chi )$ are 'near' each other. If $d$ is an algebraic irrational number then this was obtained by T. Nakamura. Ł. Pańkowski solved the case then $d$ is a transcendental number. We prove the case then $d\ne 0$ is a rational number. If $d=0$ then by B. Bagchi we know that the above hypothesis is equivalent to the Riemann hypothesis for the given Dirichlet $L$-function. We also consider a more general version of the above problem.

arXiv commentUnfortunately the proof of Theorem 1 contains a gap. The gap is partially covered in T. Nakamura and L. Pankowski, Erratum to: The generalized strong recurrence for non-zero rational parameters, Arch. Math. 99 (2012), 43-47. Theorem 2 is not affected by this gap. J. Number Theory, 131(7) (2011)