From: "James Van Buskirk"
Subject: Re: question about primes
Date: Sun, 2 Jan 2000 03:06:01 -0700
Newsgroups: sci.math
denis.feldmann wrote in message <84n5vq$eq8$1@wanadoo.fr>...
>But isn't it true that limit ("number of primes modulo 4"/ "number of primes oo)?
>What is the exact meaning of Chebyshev's Bias, then?
To get a feel for Chebyshev's bias, take the data I posted for the
congruence classes mod 10 and plot the functions
(pi(x;10,r)-(pi(x)-2)/phi(10))*ln(x)/sqrt(x)
for r = (/1,3,7,9/) on the y axis (x on x-axis) all on the same
graph. Also plot the sum of the two functions for nonresidues
and the sum of the two functions for quadratic residues on
another graph. Make the x-axis logarithmic scale. It's more striking
if you plot perhaps 100 points per decade but that would have been a
huge post. Chebyshev's bias doesn't mean that the nonresidue primes
always outnumber the residue primes (2082927199 is the first x for
which the residues are in the lead) but it does mean that they do more
often than not by perhaps O(ln(x)/sqrt(x)) or so.
==============================================================================
From: "denis.feldmann"
Subject: Re: question about primes
Date: Mon, 3 Jan 2000 16:40:08 +0100
Newsgroups: sci.math
John R Ramsden a �crit dans le message <38712cf9.717131@news.demon.co.uk>...
>GMT, jpr2718@aol.com wrote:
>>
>> Mark Adkins wrote:
>> >
>> > Can anyone tell me whether primes which end in nines and ones
>> > become rarer (relative to primes which end in sevens and threes)
>> > the further out one goes into larger integers?
>>
>> For a more subtle discussion of this phenomenon, see "Chebyshev's Bias"
>> by M. Rubinstein and P. Sarnak. This paper is available in PostScript
>> on the web, but I can't recall where. There used to be a link to this
>> paper at http://www.math.harvard.edu/~elkies/M259.98/index.html, but I
>> don't know whether that still works.
>>
>> The authors state that, "Chebyshev noted in 1853 that there are many
>> more primes congruent to 3 than 1 modulo 4." They make clear the sense
>> in which this is true, and investigate the relative abundance of primes
>> in different residue classes for a number of moduli, including 5.
>
>I had a vague recollection that Kolmogorof [sp?] proved that in the
>limit primes are equally distributed into every residue class prime
>relative to a modulus. I wouldn't bet a months salary on that though,
>and maybe someone else can clarify and/or correct it.
>
>
After some Web searches, i got the following:
1) you are right (except for the Kolmogoroff attribution; i thought it was
Siegel, but couldn't find a clear reference)
2) but *still*, there is a bias: even if lim (f(x)/g(x)=1), it would be
thought as a bias in distribution if f(x)>g(x). That is not exactly the
case, but what happens is f(x) >g(x) much more often than the opposite!
For exact results (usually depending also of generalised Riemann hypothesis,
see:
http://pauillac.inria.fr/algo/banderier/Seminar/Vardi/
(in English, happily)
[quoted sig deleted --djr]