Talk:Prime Number Theorem

new page - improve!

I started this article from material taken from Prime number. Some of what's currently in this article should probably go instead in an article about the Riemann zeta function. Hard to know how detailed to get on this page - it's serious mathematics. - Greg Martin 22:14, 29 April 2007 (CDT)

Capital letters

Perhaps the page name should be "Prime number theorem" rather than "Prime Number Theorem". --Catherine Woodgold 21:07, 30 April 2007 (CDT)

log vs ln

Shouldn't the equation be written as ln(x) or log_e(x) if it is a natural log? David E. Volk 15:02, 12 November 2008 (UTC)

David, I agree, see here --Paul Wormer 16:04, 12 November 2008 (UTC)

Euler's proof of the infinitude of primes

I would prefer to see this stated something like the following:

Euler used his formula to give an alternative proof that there are infinitely many primes. Let S be a finite set of primes and (S) the integers divisible only by primes in S. Taking $s=1$, we get

$\sum _{n\in (S)}{\frac {1}{n}}=\prod _{p\in S}\left({1-{\frac {1}{p}}}\right)^{-1},$

since the product on the right-hand side is a finite product of absolutely convergent geometric series and so may be rearranged to form the sum on the left. But that sum is therefore convergent and so cannot equal the harmonic series, which diverges. Hence (S) cannot be the set of all integers, and this shows that there are infinitely many primes.