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.
--actually, any upperbound x^(1/2+o(1)) is equivalent to RH. Therefore, if we use the ceiling(sqrt(x)) function instead of sqrt(x), and the 1+1/2+...+1/(x-1) function instead of ln(x), then the right hand side becomes entirely made of integers and still legitimate... it similarly is ok to replace the li(x) function by anything close enough to it... the point is, we are allowed to weaken the Schoenfeld inequality in any way we want, provided (1) it genuinely is a weakening, and (2) the weakened right hand side asymptotically is x^(1/2 + o(1)), and then the resulting statement will still be equivalent to RH. And of course do so in such a way as to make the statement all-integer and easy to program. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)