On Robin's criterion for the Riemann Hypothesis

Details On Robin's criterion for the Riemann Hypothesis On Robin's criterion for the Riemann Hypothesis

2006-04-13

arxiv.org/abs/math/0604314v2

J. Theor. Nombres Bordeaux 19 (2007), 351-366.

Mathematics

Number Theory

15 pages. Corrected version. In the first version in Theorem 5 (main result) it was falsely asserted that n must be superabund

Scientific paper

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d=5041, where gamma denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n>=37 does not satisfy Robin's criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power >1. As a consequence we infer that RH holds true if and only if every natural number divisible by a fifth power >1 satisfies Robin's inequality.

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