A glimpse inside the mathematical kitchen

TL;DR

This paper proves a new inequality involving alternating series with cosine terms, introduces a numerical technique for inequality proofs, and applies these results to improve bounds in the Riemann zeta function analysis.

Contribution

It presents a novel inequality proof using a numerical maximal slope principle and analytical methods, with applications to bounds in the Riemann-Siegel expansion.

Findings

Proved the inequality for 0 < r <= 1 and 0 < phi < pi.

Developed a general rearrangement theorem for inequalities.

Applied results to obtain sharp error bounds in Riemann zeta function approximations.

Abstract

We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1} < sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For the case r = 1 we give two proofs. The first one is by means of a general numerical technique (maximal slope principle) for proving inequalities between elementary functions. The second proof is fully analytical. Finally we prove a general rearrangement theorem and apply it to the remaining case 0 < r < 1. Some of these inequalities are needed for obtaining general sharp bounds for the errors committed when applying the Riemann-Siegel expansion of Riemann's zeta function.