From: johana@matematik.su.se (Johan Andersson) Newsgroups: sci.math.research Subject: Re: Strong prime number theorem wanted Date: 10 Sep 1998 13:03:55 GMT [Posted and mailed] In article <35F4544D.D3E4CB43@daimi.aau.dk>, Rasmus Pagh writes: > Hi, > > I'm looking for a theorem of the form: > > "There is always a prime between n and n+f(n)" > > For f(n)=n, it's a classic result. From a result in > A.E.Ingham's 1932 book "The distribution of prime numbers", > the theorem can be derived with f(n)=O(n log n exp(-c sqrt(log n))) > for some c>0. > > Since it has been 66 years, I would like to ask what is the best > (slowest growing) function f currently known? > > /Rasmus Harman-Baker (Baker, R. C.; Harman, G. The difference between consecutive primes. Proc. London Math. Soc. (3) 72 (1996), no. 2, 261--280.) have shown that you can take f(n)=n^0.535 (for n>n_0) If you want the true local asymptotic density of the primes you can use zero-density estimates of the Riemann zeta-function to show that you can get pi(x+h)-pi(x) ~ h/(log x), where pi(x) denotes the number of primes below x, and h>(log X)^N x^{7/12}. Heath-Brown has used sieve-methods to show that this can be improved to x^{7/12-w(x)}OO} w(x)=0. The Riemann hypothesis as well as the much weaker density hypothesis (That is that N(sigma,T)=O(T^{2(1-sigma)+epsilon}, where N(sigma,T) denote the number of zeroes of the Riemann zeta-function with real part less than sigma and absolute value of imaginary part less than T) does not give anything much better than f(n)=n^{1/2+epsilon} (The n^epsilon can at best be replaced by a log-power). Of course these results are rather week, since we of expect that we can chose f(n)=n^epsilon or even f(n) a log-power. Cramer thought that we can chose f(n)=(log n)^2. If we want to use zeta-function theoretic methods we need very strong results on the zeroes of the Riemann zeta-function to obtain those types of results. We do not only need the Riemann hypothesis. We also need some strong cancellation when we sum over the zeroes in the Weil explicit formula. We cannot just use absolute values. Johan Andersson Department of Mathematics Stockholm University