From: bobs@mathworks.com (Bob Silverman) Newsgroups: sci.math Subject: Re: Ratio of successive primes? Date: 12 Sep 1995 19:13:32 -0400 In article <435360$hen@vixen.cso.uiuc.edu>, Charles Blair wrote: > > Define f(epsilon) to be the smallest prime p such that >p divided by the next-smaller prime is less than 1+epsilon. > > What is known about f(epsilon)? I am not interested in >best possible results so much as something easy to cite--- >I am working on a (non-number theory) construction that would >involve facts about f(epsilon). > >Charles Blair f(epsilon) will be a very irregular function. Prime gaps just vary too much. Check the following: Robin, G. Estimation del la fonction de Tchebychev sur le k'ieme nombre premier .......Acta Arith, 42, 1983 pp. 367-389 For example (based on Riemann zero calculations) these are the tightest know inequalities of their type: if n > 2, p_n > nlog n + n(loglog n - 1.0072629) if n > 2 and p_n < 10^11 then p_n > n log n + n(loglog n - 1) if n > 7022 then p_n < n log n + n(log log n - .9385) Cramer's conjecture (now believe wrong) says there is always a prime between n and n + log^2 n. The exact true nature of this conjecture is not known. It may require n + (log n)^(2 + e) for e > 0 or replacing e by o(1) may be good enough. Noone really knows. The best PROVED result is that there is always a prime between n and n + n^(11/20 - 1/384 + e) for some e > 0. -- Bob Silverman The MathWorks Inc. 24 Prime Park Way Natick, MA