Let

denote the sum of the divisors of .

G. Robin [
MR 86f:11069] showed that the Riemann Hypothesis is
equivalent to

for all , where is Euler's constant. That inequality does not leave much to spare, for Gronwall showed

and Robin showed unconditionally that

for .

J. Lagarias [
arXiv:math.NT/0008177] elaborated on Robin's
work and showed that the Riemann Hypothesis is equivalent to

for all , where is the harmonic number

By definition,

so Lagarias' and Robin's inequalities are the same to leading order.

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