NetNews Usenet Archive 1993 #3/NN_1993_3.iso/spool/sci/math/18729

Text File | 1993-01-25 | 2.0 KB | 41 lines

Newsgroups: sci.math
Path: sparky!uunet!pipex!warwick!pavo.csi.cam.ac.uk!camcus!cet1
From: [e-mail] (C.E. Thompson)
Subject: Re: Distribution of primes mod 4
Message-ID: <[e-mail]>
Sender: [e-mail] (USENET news)
Nntp-Posting-Host: bootes.cus.cam.ac.uk
Organization: U of Cambridge, England
References: <[e-mail]> <[e-mail]> <[e-mail]> <[e-mail]>
Date: Mon, 25 Jan 1993 15:47:09 GMT
Lines: 28

In article <[e-mail]>, [e-mail]
(Noam Elkies) writes:

[Many thanks to Noam for sorting out the confusion rampant in previous postings]

|>                                                Now in the Prime
|> Number Theorem, when we pass from the sum of \Lambda(m) to the
|> count of primes <x the first thing we do is ignore the contribution
|> of all prime powers higher than the first, which contribute
|> "negligibly" to the sum.  However in our case this "negligible"
|> contribution is asymptotically sqrt(x) [note that chi(p^2) is
|> always 1 for p>2!].  Thus we find that pi(x;1,4)-pi(x;3,4)
|> is approximated by sqrt(x)/log(x) times an almost periodic
|> function of log(x) whose mean is -1!

Maybe it is worth pointing out that the situation isn't all that different
from PNT (estimating pi(x)=pi(x;0,1)). The function which is "centred" on
li(x) is not pi(x), but \sum_{n<=x}{\Lambda_1(n)} = pi(x) + (1/2)pi(x^{1/2}}
+ (1/3)pi{x^{1/3)} + ... Again the oscillations can become larger than any
constant multiple of sqrt(x)/log(x), so that pi(x) > li(x) sometimes
(although not for any explicitly computed x, as yet). One expects (given
RH + suitable independence constraints on the zeroes of the zeta function)
that the set of x such that pi(x) > li(x) has a non-zero (rather small)
limiting *logarithmic* density, but no limiting natural density.

Chris Thompson
JANET:    [e-mail]
Internet: [e-mail]