A glimpse inside the mathematical kitchen

Details A glimpse inside the mathematical kitchen A glimpse inside the mathematical kitchen

2010-04-03

arxiv.org/abs/1004.0469v1

Mathematics

Classical Analysis and ODEs

14 pages, 2 figures

Scientific paper

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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