Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis

Details Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis

2011-10-23

arxiv.org/abs/1110.5078v2

Integers 11 (2011) article A33

Mathematics

Number Theory

11 pages, 1 table, clarified Proposition 4, added reference 4

Scientific paper

For n>1, let G(n)=\sigma(n)/(n log log n), where \sigma(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime factors p of N and all multiples aN of N. The proof uses Robin's and Gronwall's theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erd\H{o}s's results.

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