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A NEW UPPER BOUND FOR THE SUM OF DIVISORS FUNCTION

Published online by Cambridge University Press:  14 August 2017

CHRISTIAN AXLER*
Affiliation:
Institute of Mathematics, Heinrich Heine University, Duesseldorf, 40225 Duesseldorf, Germany email christian.axler@hhu.de

Abstract

Robin’s criterion states that the Riemann hypothesis is true if and only if $\unicode[STIX]{x1D70E}(n)<e^{\unicode[STIX]{x1D6FE}}n\log \log n$ for every positive integer $n\geq 5041$ . In this paper we establish a new unconditional upper bound for the sum of divisors function, which improves the current best unconditional estimate given by Robin. For this purpose, we use a precise approximation for Chebyshev’s $\unicode[STIX]{x1D717}$ -function.

Type
Research Article
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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