Re: Pi and the distribution of prime numbers

From: Jim Spriggs (jim.sprigs_at_ANTISPAMbtinternet.com.invalid)
Date: 02/09/05 

Date: Wed, 9 Feb 2005 22:43:02 +0000 (UTC)

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)?

Well, I don't know about _distribution_ of the primes, but one proof
that the sum of the reciprocals of all prime numbers diverges to
infinity uses

   log log n - log(pi^2/6) < sum_{p <= n} p^{-1}

where the sum is over primes.

It would be nice to know a small blah so that

   log log n - log(pi^2/6) < sum_{p <= n} p^{-1} < blah.

Relevant Pages

  • Re: Mr. P and Ms. S
    ... That means there are only two ways to factor xy out of which one is sum of two primes and another is not sum of primes and he knows the later is the way. ... That is all possible decomposition of x+y there is only one decomposition which yields only two ways of factoring out of which one can be written as sum of two primes. ... Sujit Gujar. ...
    (sci.math)
  • Re: approaching a proof
    ... > number is the sum of two primes). ... > probabilistic sense, the limit of our proofs as M goes to infinity ...
    (sci.logic)
  • Quantum Gravity 313.3: More Regarding Exp[ln(p1) + ln(p2) + ... + ln(pn)] As Fundamentally Additive
    ... powers of primes), is in a sense "irreducible" and the sum ln+ ln ... In the well known decomposition: ... except for a function exp (the exponential function). ...
    (sci.physics)
  • Re: Mr. P and Ms. S
    ... of which one is sum of two primes and another is not ... of x+y there is only one decomposition which yields ... yeild more than 2 ways of factoring. ...
    (sci.math)
  • Re: Some question in prime number!
    ... He didn't say "a sum of two distinct primes", ... every positive integer can be written as a product of primes, ...
    (sci.math)