omega-Theorems for Quotients of Zeta-Functions at Combinations of Points

Theorem A. Let q >/= O and r >/= O be integers. Let s = sigma + it, let zeta(s) be the Riemann zeta-function, let G(o)(s) = 1, and [Formula: see text] and let F(s) = G(q)(s)/H(r)(s). Then as t --> infinity lim sup [unk]F(1 + it)[unk]/(log log t)(q+r+1) >/= (6/pi)(2))(r+1) exp {(q + r + 1)gamma}, where gamma is Euler's constant.Stronger results such as proved in [1] are valid, and in particular q and r can be allowed to increase with t as in [1]. Results involving the real part of the sum of the factors of G(q) and of the reciprocals of the factors of H(r) can be proved much as in [3].