From: cet1@cus.cam.ac.uk (Chris Thompson)
Subject: Re: "The Big Guns of Math" & Unsolved problems
Date: 12 Sep 1999 12:44:09 GMT
Newsgroups: sci.math
Keywords: Locations of zeros of zeta

In article <37d12eeb.2578573@news.demon.co.uk>,
John R Ramsden <jr@redmink.demon.co.uk> wrote:
>On Wed, 01 Sep 1999 11:23:45 -0500, Stephen Montgomery-Smith
><stephen@math.missouri.edu> wrote:
>>
>> I know for a fact that Hardy worked on the Riemann zeta
>> conjecture - solving it was one of his stated ambitions, along 
>> with murdering Mussolini, proving the non-existence of God, and
>> making a winning play in an important game of cricket.  (He 
>> failed at all three.)
>
>He may not have cracked the Riemann Hypothesis completely, but he
>had it worried though. Didn't he prove that at least a certain
>proportion (1/5?) of the roots must lie on the critical line?
>I think that value has been increased since.

Hardy proved any number of results revelant to the Riemann Hypothesis,
but not this particular one.

Let N_0(T) be the number of zeros on the criticial line with imaginary
part in [0,T], to be compared with the total number of zeros with such
imaginary parts, which is N(T) = (T/2pi)log(T/2pi) - (T/2pi) + O(log T).
Then 

  N_0(T) -> infinity           was proved by Hardy (1914)
  N_0(T) > cT for T>k          was proved by Hardy & Littlewood (1921)
  N_0(T) > c'T log(T) for T>k' was proved by Selberg (1942)

Full references (and the proofs) are in Titchmarsh's book.

The c' given by Selberg's original proof was extravagantly small. I think
that it has indeed been much improved since then, but I don't have
references to hand.

Chris Thompson
Email: cet1@cam.ac.uk
==============================================================================

From: gerry@mpce.mq.edu.au (Gerry Myerson)
Subject: Re: "The Big Guns of Math" & Unsolved problems
Date: Tue, 14 Sep 1999 16:15:51 +1100
Newsgroups: sci.math

In article <7rg76p$cl4$1@pegasus.csx.cam.ac.uk>, cet1@cus.cam.ac.uk (Chris
Thompson) wrote:

=> Let N_0(T) be the number of zeros on the criticial line with imaginary
=> part in [0,T], to be compared with the total number of zeros with such
=> imaginary parts, which is N(T) = (T/2pi)log(T/2pi) - (T/2pi) + O(log T).
=> Then 
=> 
=>   N_0(T) -> infinity           was proved by Hardy (1914)
=>   N_0(T) > cT for T>k          was proved by Hardy & Littlewood (1921)
=>   N_0(T) > c'T log(T) for T>k' was proved by Selberg (1942)
=> 
=> Full references (and the proofs) are in Titchmarsh's book.
=> 
=> The c' given by Selberg's original proof was extravagantly small. I think
=> that it has indeed been much improved since then, but I don't have
=> references to hand.

Levinson got 1/3, then in 1989 Conrey got 2/5. Crelle 399 (1989) 1-26, 
MR 90g:11120. 

Gerry Myerson (gerry@mpce.mq.edu.au)
==============================================================================

From: Raymond Manzoni <raymman@club-internet.fr>
Subject: Re: Riemann Perturbances
Date: Tue, 18 May 1999 21:55:21 +0200
Newsgroups: sci.math

F.Baube(tm) wrote:

> Can anyone direct me to a web page that tabulates the
> first nontrivial zeroes of the Riemann zeta function ?

(snip)


    Hi,

    Concerning your first question. Have a look at the following site
which will perhaps give you more than you hoped !
http://www.lacim.uqam.ca/piDATA/zeta100.html

    and this one too:
    http://www.research.att.com/~amo/zeta_tables/index.html


    Hope it partially helped,

                    Raymond Manzoni