Báez-Duarte's Criterion for the Riemann Hypothesis and Rice's Integrals

2006/03/30 by Krzysztof Maslanka, Maslanka, Krzysztof
Mathematics · #11M06 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11M06

paper · pdf · doi:10.48550/arxiv.math/0603713

10 pages, 6 figures

arxiv created 2006/04/01 · arxiv updated 2009/12/01

Abstract

Criterion for the Riemann hypothesis found by Báez-Duarte involves certain real coefficients c_kdefined as alternating binomial sums. These coefficients can be effectively investigated using Nö% rlund-Rice's integrals. Their behavior exhibits characteristic trend, due to trivial zeros of zeta, and fading oscillations, due to complex zeros. This method enables to calculate numerical values of c_kfor large values of k, at least to k=4⋅ 10⁸. We give explicit expressions both for the trend and for the oscillations. The first tends to zero and is therefore, in view of the criterion, irrelevant for the Riemann hypothesis. The oscillations can be further decomposed into a series of harmonics with amplitudes diminishing quickly. Possible violation of the Riemann hypothesis would indicate that the amplitude of some high harmonic increases.

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