360pan gold
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 Posted 22/05/2007 04:56:40 PM | | Subject: Re: Strong prime number theorem
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)}<h<x, if lim_{h->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
360
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