Generalization of Recent Method Giving Lower Bound for N(o)(T) of Riemann's Zeta-Function

Let h(s) = pi(-s/2)tau(s/2). Then, h(s)zeta(s) approximately h(s)H(s) + h(1 - s)H(1 - s) where H(s) = Sigma(1 - (log n)/log t/2pi)n(-s), n /= N(T)/3. Here the extension to H(s) approximately Sigma P (1 - (log n)/log t/2pi) 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.