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arXiv · 1004.0469

A glimpse inside the mathematical kitchen

Juan Arias-de-Reyna·Jan van de Lune
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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.

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Juan Arias-de-Reyna, Jan van de Lune. 2010-04-03. A glimpse inside the mathematical kitchen. https://doi.org/10.7153/jmi-05-30

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