The Nicolas and Robin inequalities with sums of two squares |
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Authors: | William D Banks Derrick N Hart Pieter Moree C Wesley Nevans |
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Institution: | (1) Department of Mathematics, University of Missouri, Columbia, MO 65211, USA;(2) Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany |
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Abstract: | In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e
γ
n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a
2 + b
2 that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e
γ
log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.
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Keywords: | Nicolas inequality Robin inequality Sums of two squares |
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