The Riemann Hypothesis, often declared to be the number one outstanding unsolved question in mathematics, usually takes some explaining to describe accurately to a layperson.

I like the equivalence, recently proved by Jeffrey Lagarias of AT&T Research, of the Riemann Hypothesis to the following Statement:

Statement:
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For any n = 1,2,3,... let sigma(n) = the sum-of-divisors function (so that for example sigma(6) = 12), and let H_n = 1 + 1/2 + 1/3 + ... + 1/n be the nth harmonic number. Let log denote log_e.

Then for all n = 1,2,3,...,

(*)   sigma(n)  <=  H_n  +  exp(H_n) * log(H_n)
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Lagarias adds the words "with equality only for n = 1".  I suspect it's easy to prove that for n > 1 the RHS of (*) is never an integer, but maybe a number theorist out there can confirm this.

--Dan A.