From: Bob Silverman <bobs@rsa.com>
Subject: Re: a question on the distribution of prime numbers
Date: Mon, 08 Nov 1999 16:37:13 GMT
Newsgroups: sci.math

In article <8062j2$iql$1@nnrp1.deja.com>,
  David Bernier <bernier@my-deja.com> wrote:
> I have heard of the unresolved conjecture:
> <<For every integer n>0, there is a prime between
>   n and (n+1)^2 .>>
>
> Suppose that, for an \eps >0 , we let:
>
> Prop( \eps) say: << For all sufficiently large
>            positive integers n, there is
>            at least one prime between n and n + n^{\eps} .>>
>
> (a) For what values of \eps is Prop(.) known to
>     be true?   (I think the case \eps = 1 is implied
>     by Bertrand's Postulate, which I think is proven)

I know that \eps = 11/20 has been fully proved. I think this may
have been improved recently. R.H. would allow one to take \eps = 1/2 + o
(1).

>
> (b) For what values of \eps is Prop(.) conjectured
>     to be true?

See above,

>
> (c) For what values of \eps is Prop(.) known to
>     be false?

None.  CRAMER's conjecture states that for all sufficiently large x
there is a prime between  n  and  n + log^2(n).   log^2(n) = n^o(1).

Recent work by Hildebrand and Maier suggests that Cramer's conj.
may not be quite correct.  Suppose the following:  there is always
a prime between  n and n + f(n).  Noone really knows what f(n) really
is,  (with or without proof).   The Hildebrand/Maier work suggests
that f(n) = log^(2 + \eps) n   but NOT log^(2 + o(1)) may be correct.
But noone really knows.

>
> (d) For what values of \eps is Prop(.) conjectured
>     to be false?

It is known that f(n) = log(2 - \eps) n can not be correct.


--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"


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From: dmoews@xraysgi.ims.uconn.edu (David Moews)
Subject: Re: a question on the distribution of prime numbers
Date: 8 Nov 1999 12:49:00 -0800
Newsgroups: sci.math

In article <8062j2$iql$1@nnrp1.deja.com>,
David Bernier  <bernier@my-deja.com> wrote:
|Suppose that, for an \eps >0 , we let:
|
|Prop( \eps) say: << For all sufficiently large
|           positive integers n, there is
|           at least one prime between n and n + n^{\eps} .>>
|
|(a) For what values of \eps is Prop(.) known to
|    be true?   (I think the case \eps = 1 is implied
|    by Bertrand's Postulate, which I think is proven)

\eps >= 0.535 (Baker & Harman, Proc. London Math. Soc. (3) 72 (1996), 261-280.)
Given the Riemann Hypothesis, it is true for all \eps > 1/2.

|(b) For what values of \eps is Prop(.) conjectured
|    to be true?

All positive \eps.

|(c) For what values of \eps is Prop(.) known to
|    be false?
|
|(d) For what values of \eps is Prop(.) conjectured
|    to be false?

No positive \eps.

It is conjectured that for each eps>0, there is always a prime between 
n and n + (log n)^(2+eps), for sufficiently large n.  

For all eps>0 and arbitrarily large n, there need not be a prime between n and

 n + (2 e^gamma - eps) (log n log log n log log log log n / (log log log n)^2)

(Pintz, J. Number Theory 63 (1997), 286-301.)
-- 
David Moews                                   dmoews@xraysgi.ims.uconn.edu
==============================================================================

From: Bob Riley <bob@math.binghamton.edu>
Subject: Re: Density.
Date: 13 May 1999 14:01:42 -0500
Newsgroups: sci.math.research
Keywords: How far from one prime to the next?

Gerry Myerson <gerry@mpce.mq.edu.au> wrote:
> In article <3739e757.0@bingnews.binghamton.edu>, Bob Riley
> <bob@math.binghamton.edu> wrote:

>>    For large primes p the next prime q is < p + p^t, where t < 1, eg
>> any t > 5/12 works.  Then log(q) = log(p) + p^(t-1) + small, so the
>> spacing between logs of consecutive primes -> 0.

> I don't think the 5/12 is right. I think you can get 1/2 from the 
> Riemann Hypothesis - but first you have to prove the Riemann Hypothesis. 

   The 5/12 isn't right, in the sense of optimal, but it is the main new
result on the topic in "The distribution of prime numbers" by MN Huxley,
Oxford Monograph.  I recall a smaller number is now known but I don't
remember what.  The 1/2 is sort of a holy grail (short of RH): is the
next prime < the current one + its square root?????  Cramer's conject-
ure from 1936 has next prime < current + (log(current))^2.

   I don't recommend Huxley: far too many misprints, and *lots* of extra
detail needs to be supplied.  Great quotations from Winnie-the-Pooh
though.

                                    R^2
==============================================================================

From: Bruno Langlois <blang@club-internet.fr>
Subject: Re: Density.
Date: Thu, 13 May 1999 09:20:23 +0200
Newsgroups: sci.math.research

> >    For large primes p the next prime q is < p + p^t, where t < 1, eg
> > any t > 5/12 works.  Then log(q) = log(p) + p^(t-1) + small, so the
> > spacing between logs of consecutive primes -> 0.
> 
> I don't think the 5/12 is right. I think you can get 1/2 from the
> Riemann Hypothesis - but first you have to prove the Riemann Hypothesis.

Ok, I received a reference :

M.Huxley, An application of the Fouvry-Iwaniec theorem,
Acta Arith. 43(1984) 441-443.

Huxley has shown that : p_(n+1)<p_n+(p_n)^(7/12+small)
So p_(n+1) ~ p_n and (log p_n) is dense in R/Z.

B.L.
==============================================================================

From: Kurt Foster <kfoster@rmi.net>
Subject: Re: (p_{k+1}/p_k)^k limit
Date: Fri, 15 Oct 1999 13:37:46 GMT
Newsgroups: sci.math

In <38064BF3.CCB1EC85@lucent.com>, Wenyi Feng <wenyi@lucent.com> said:
 Leroy Quet wrote:

.. Let p_k be the k_th prime.
.. What is the limit, as k->infinity, of [(p_{k+1}/p_k)^k]?

. Guess it is e^2.

  I don't think there is a limit.  For, calling the proposed limit L, we
have

L = LIMIT, k -> infinity, (1 + (p_(k+1) - p_k)/p_k)^k

ln(L) = LIMIT, k -> infinity, k * (p_(k+1) - p_k)/p_k

  Now p_k/(k * ln(k)) -> 1 by PNT whence ln(k)/ln(p_k) -> 1 also.  Thus,

ln(L) = LIMIT, k -> infinity, (p_(k+1) - p_k)/ln(p_k).

  Unfortunately, this quotient has no limit.  It is known (proved
independently by Erdos and Ricci) that the limit points form a set of
positive Lebesgue measure, though no specific number has been identified
as a limt point!  It is known (Rankin) that the lim sup is infinite, and
(Bombieri and Davenport) the lim inf is =< 0.46.  If the Twin Primes
Conjecture is true, then the lim inf is 0.  So lim inf L is known to be
exp(0.46) at most, ans lim sup L is infinite.