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]