Re: Pi and the distribution of prime numbers
From: Bart Goddard (goddardbe_at_netscape.net)
Date: 02/10/05
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Date: 10 Feb 2005 21:24:11 GMT
José Carlos Santos wrote:
> Hi all,
>
> At the MathWorld page dedicated to pi, located at
>
> http://mathworld.wolfram.com/Pi.html
>
> it is claimed that pi "crops up in all sorts of unexpected places in
> mathematics [...]. For example, it occurs in [...] the distribution of
> primes". Does anyone know what's the connection? Or is it a confusion
> between the number pi and the prime counting function (usually denoted
> by pi)?
I would guess that the connection they're talking about here
is the fact that \pi appears in the functional equation for
the zeta function. It appears there because it's dragged
in by the gamma function. We expect to see \pi when we're
doing contour integrals and sqrt{\pi} when we're integrating
exponentials.
Bart
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