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 < σ < 1 , and let L ( s , χ ) be the Dirichlet L-function. The hypothesis is that for any real number d there exist ‘many’ real numbers τ such that the shifts L ( s + i τ , χ ) and L ( s + i d τ , χ ) 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 ≠ 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.