On Dirichlet series with identical beginnings

It is shown that if Vinogradov's conjecture is false for a Dirichlet character (mod q), then ζ(s) and L(s) are very similar in regions of the critical strip where ζ(s), L(s) are small. In particular, ζ(s) = L(s + h(s)) (where h(s) → 0) in such regions.