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

Theorem A. Let q ≥ O and r ≥ O be integers. Let s = σ + it, let ζ (s) be the Riemann zetafunction, let Go(s) = 1, and Gq(s) = ζ (2s - 1)ζ (4s - 3) ... ζ (2qs - 2q + 1), q ≥ 1 Hr(s) = ζ (s)ζ (3s - 2) ... ζ ((2r + 1)s - 2r) and let F(s) = Gq(s)/Hr(s). Then as t → ∞ lim sup |F(1 + it)|/(log log t)q+r+1≥ (6/π )2)r+1exp {(q + r + 1)γ }, where γ 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 Gqand of the reciprocals of the factors of Hrcan be proved much as in [3].