From: shallit@graceland.uwaterloo.ca (Jeffrey Shallit)
Newsgroups: sci.math.research
Subject: Re: Request: Reference for prime-number result.
Date: Mon, 11 Mar 1996 14:17:06 GMT

In article <4hq0ib$aov@nyx10.cs.du.edu>, Shrisha Rao <shrao@nyx.net> wrote:
>Greetings.
>
>I am looking for a proof showing that for any n in the set of positive
>integers, there always exists a prime number between n and 2n. I was
>told that a tighter bound has been shown, in fact, that there is
>always a prime between n and n + n^(3/4). However, I have not been
>able to come across any references to this result.
>
>Any information about such references, if possible also indicating
>what the currently known best bound for the confirmed presence of a
>prime between n and some function of n is, would be greatly
>appreciated.
>
>Regards,
>
>Shrisha Rao

Ivic's book, _The Riemann Zeta-Function_ (Wiley, 1985) contains a
summary of what was known until 1985.

The best result in this area known to me is the inequality
	
	p_n - p_{n-1} = O(p_n^a)

where a = 1/2 + 7/200, due to Baker and Harman.  It is cited
in Andrew Granville's technical report 8, "Harald Cramer and
the distribution of prime numbers", University of Georgia,
Department of Mathematics, 1994.  

Shameless self-plug: much more information on these kinds of
questions can be found in my forthcoming book with Eric Bach,
_Algorithmic Number Theory_, to be published in the next few
months by MIT Press.  My home page contains a table of contents.

Jeffrey Shallit, Computer Science, University of Waterloo,
Waterloo, Ontario  N2L 3G1 Canada shallit@graceland.uwaterloo.ca
URL = http://math.uwaterloo.ca/~shallit/

==============================================================================

Newsgroups: sci.math
From: rdsilverman@qed.com
Subject: Re: For every prime p, is there a prime between p and 2p?
Date: Tue, 12 Mar 96 09:35:41 PDT

In article <4i1tsp$f1k@dartvax.dartmouth.edu>, <Benjamin.J.Tilly@dartmouth.edu> writes:
Stuff deleted...


> >         Also, for any epsilon > 0 there is an N so that if x > N then there is
> >    a prime between x and x(1 + epsilon).  So you can improve Bertrand as much
> >    as you want, provided you are willing to take a walk up to the relevant N
> >    and start there.  It is the _small_ x's that are tough.
> 
> I remember looking at this years ago and not finding very good
> bounds...but are there bounds or conjectures today on how fast epsilon
> decreases as N increases? (I had a table at one point going out to n


There are rather sharp bounds that are known. Check out the paper in
Math. Comp. (it was quite a few years ago) by Rosser and Schoenfeld.
Their results have been further sharpened by Guy & Robin.

The bounds were established by extensive compution of Zeta function zeros.

I'm a little surprised that you don't know of this paper, Ben. "Rosser
and Schoenfeld" is a very well known and often quoted paper.

==============================================================================

From: rgep@newton.cam.ac.uk (R.G.E. Pinch)
Newsgroups: sci.math.research
Subject: Re: Request: Reference for prime-number result.
Date: 12 Mar 1996 14:28:19 GMT
Summary: There is a prime between x a and x + x^{23/42}
Keywords: prime, bound, reference

In article <4hq0ib$aov@nyx10.cs.du.edu> 
shrao@nyx.net (Shrisha Rao) writes:
>I am looking for a proof showing that for any n in the set of positive
>integers, there always exists a prime number between n and 2n. I was
>told that a tighter bound has been shown, in fact, that there is
>always a prime between n and n + n^(3/4). 
>Any information about such references, if possible also indicating
>what the currently known best bound for the confirmed presence of a
>prime between n and some function of n is, would be greatly appreciated.

According to A. Ivic, The Riemann zeta-function, Wiley, 1985, one
can prove that for y = x^t, t > 13/23 = 0.56521..., there are at
least y / (177 log x) primes between x-y and y, and Iwaniec and Pintz
show that there is a prime between x and x + x^t for t > 23/42 = 0.54761...
The Riemann hypothesis would imply a prime between x and x^(1/2).log x
There is a useful table of previous results in this direction.

Richard Pinch;	Queens' College, Cambridge