Newsgroups: sci.math Path: hermes.uni-sb.de!sbusol.rz.uni-sb.de!news.coli.uni-sb.de!news.dfn.de!darwin.sura.net!dcc.uchile.cl!cedro!ccea From: ccea@cec.uchile.cl (CEA BASTIDAS CHRISTIAN) Subject: ln(2)+ln(3)+ln(5)+ln(7)+...+ln(p) Message-ID: <D2x0tD.4AM@dcc.uchile.cl> Sender: usenet@dcc.uchile.cl (News) Organization: Universidad de Chile, Depto. de Ciencias de la Computacion X-Newsreader: TIN [version 1.2 PL2] Date: Tue, 24 Jan 1995 15:10:25 GMT Lines: 63 [ Article crossposted from sci.math ] [ Author was Robert J. Harley ] [ Posted on 20 Jan 1995 14:44:37 GMT ] caldwell@unix1.utm.edu (Chris Caldwell) writes: >I am looking for a pratical, accurate approximation of > > ln(p#) = ln(2)+ln(3)+...+ln(p) > >(where the sum is taken over the primes <= p). My values >of p are about 10^10. Any suggestion? How accurate do you need? You could use ln(p#) ~ p. Let q denote primes and pi() the prime counting function: ln(p#) = SUM_{q <= p} ln(q) <= SUM_{q <= p} ln(p) = pi(p).ln(p) ~= p since pi(x) ~ x/ln(x) Conversely, fix c >= 1: ln(p#) = SUM_{q <= p} ln(q) >= SUM_{p/c < q <= p} ln(q) >= SUM_{p/c < q <= p} ln(p/c) = (pi(p)-pi(p/c)).ln(p/c) ~ (p/ln(p)-p/(c.ln(p/c))).ln(p/c) = p(1-ln(c)/ln(p)-1/c) ~ p(1-1/c) And c can be chosen arbitrarily large. The approximation ln(p#) ~ p is not too bad even for "small" p.