Self-approximation of Dirichlet L-functions

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. L Pankowski solved the case then d is a transcendental number. We prove the case then d not equal 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. (C) 2011 Elsevier Inc. All rights reserved.