The Riemann hypothesis is equivalent to the boundedness of the normalized divisor sum sigma_{1/2}(n)/(n^{1/2} exp(li((log n)^{1/2}))).
Broughan, Equivalents of the Riemann hypothesis
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Maximal order for divisor functions and zeros of the Riemann zeta-function
The Riemann hypothesis is equivalent to the boundedness of the normalized divisor sum sigma_{1/2}(n)/(n^{1/2} exp(li((log n)^{1/2}))).