(fwd) ln(2)+ln(3)+ln(5)+ln(7)+...+ln(p)

Patrick Schmeer (pasc@stud.uni-sb.de)
Thu, 26 Jan 95 14:24:19 +0100

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From: ccea@cec.uchile.cl (CEA BASTIDAS CHRISTIAN)
Subject: ln(2)+ln(3)+ln(5)+ln(7)+...+ln(p)
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Date: Tue, 24 Jan 1995 15:10:25 GMT
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[ 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.