Irrationality measure and lower bounds for pi(x)

Details Irrationality measure and lower bounds for pi(x) Irrationality measure and lower bounds for pi(x)

2007-09-13

arxiv.org/abs/0709.2184v3

Mathematics

Number Theory

Version 3.0, 6 pages. E. Kowalski and T. Rivoal both pointed out that the irrationality bound on zeta(2) uses the Prime Number

Scientific paper

Using the fact that the irrationality measure of zeta(2) = pi^2/6 is finite, one can deduce explicit lower bounds for the number of primes at most x. The best estimate this method yields is (basically) a lower bound of loglog(x) / logloglog(x) for infinitely many x, almost as good as Euclid's argument. Unfortunately, the standard proofs of the finiteness of the irrationality measure of zeta(2) use the prime number theorem to estimate lcm(1,...,n)! By a careful analysis of the irrationality measure constructions, we not only remove this assumption (for our applications), but prove that if g(x) is any function which is o(x/log x), then for infinitely many x we have pi(x) > g(x).

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