http://arxiv.org/abs/math/0008177 Somewhat annoyingly, Lagarias uses the exp(x) and ln(x) functions, so his reformulation of the Riemann hypothesis is not entirely about integer arithmetic; real number arithmetic also is needed. Which kind of makes you wonder why he bothered, since as everybody already knew, the RH is equivalent to an error bound for the prime-counting function expressible using some real-functions such as Li(x), e.g. see Lowell Schoenfeld: Maths of Comput 30,134 (April 1976) 337-360, e.g: |PrimeCountingFn(x) - li(x)| < sqrt(x) * ln(x) / (8*pi) for x>2657 <==> RH. An entirely integer statement was cooked up by Martin Davis, Yuri Matijasevic, and Julia Robinson: Hilbert's Tenth Problem. Diophantine Equations: Positive Aspects of a Negative Solution, pp/ 323-378 in "Mathematical developments arising from Hilbert problems", Proceedings of Symposium of Pure Mathematics, XXVIII AMS 1974. See p.335. a collection: https://web.archive.org/web/20120731034246/http://aimath.org/pl/rhequivalenc... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)