Generalization of Recent Method Giving Lower Bound for No(T) of Riemann's Zeta-Function

摘要

Let h(s) = π-s/2τ(s/2). Then, h(s)ζ(s) ∼ h(s)H(s) + h(1 - s)H(1 - s) where H(s) = Σ(1 - (log n)/log t/2π)n-s, n ≤ t/2π, led to No(T) ≥ N(T)/3. Here the extension to H(s) ∼ Σ P (1 - (log n)/log t/2π) n-s is made where P(x) is a polynomial such that P(0) = 0 and P(x) + P(1 - x) = 1. The earlier case is P(x) = x. The relevant formulas in the general case can be obtained explicitly by the earlier method used for P(x) = x, and, indeed, in some respects there is greater simplicity for the general case.