Editing Prime number theorem (section)

=== Non-vanishing on Re(''s'') = 1 ===

To do this, we take for granted that {{math|''ζ''(''s'')}} is [[Meromorphic function|meromorphic]] in the half-plane {{math|Re(''s'') > 0}}, and is analytic there except for a simple pole at {{math|''s'' {{=}} 1}}, and that there is a product formula
: <math>\zeta(s)=\prod_p\frac{1}{1-p^{-s}} </math>
for {{math|Re(''s'') > 1}}. This product formula follows from the existence of unique prime factorization of integers, and shows that {{math|''ζ''(''s'')}} is never zero in this region, so that its logarithm is defined there and
: <math>\log\zeta(s)=-\sum_p\log \left(1-p^{-s} \right)=\sum_{p,n}\frac{p^{-ns}}{n} \; .</math>

Write {{math|''s'' {{=}} ''x'' + ''iy''}} ; then
: <math>\big| \zeta(x+iy) \big| = \exp\left( \sum_{n,p} \frac{\cos ny\log p}{np^{nx}} \right) \; .</math>

Now observe the identity
: <math> 3 + 4 \cos \phi+ \cos 2 \phi = 2 ( 1 + \cos \phi )^2\ge 0 \; ,</math>
so that
: <math>\left| \zeta(x)^3 \zeta(x+iy)^4 \zeta(x+2iy) \right| = \exp\left( \sum_{n,p} \frac{3 + 4 \cos(ny\log p) + \cos( 2 n y \log p )}{np^{nx}} \right) \ge 1</math>
for all {{math|''x'' > 1}}. Suppose now that {{math|''ζ''(1 + ''iy'') {{=}} 0}}. Certainly {{mvar|y}} is not zero, since {{math|''ζ''(''s'')}} has a simple pole at {{math|''s'' {{=}} 1}}. Suppose that {{math|''x'' > 1}} and let {{mvar|x}} tend to 1 from above. Since <math>\zeta(s)</math> has a simple pole at {{math|''s'' {{=}} 1}} and {{math|''ζ''(''x'' + 2''iy'')}} stays analytic, the left hand side in the previous inequality tends to 0, a contradiction.

Finally, we can conclude that the PNT is heuristically true. To rigorously complete the proof there are still serious technicalities to overcome, due to the fact that the summation over zeta zeros in the explicit formula for {{math|''ψ''(''x'')}} does not converge absolutely but only conditionally and in a "principal value" sense. There are several ways around this problem but many of them require rather delicate complex-analytic estimates. Edwards's book<ref>{{cite book |last = Edwards |first = Harold M. |author-link = Harold Edwards (mathematician) |title = Riemann's zeta function |publisher = Courier Dover Publications |year = 2001 |isbn = 978-0-486-41740-0}}</ref> provides the details. Another method is to use [[Ikehara's Tauberian theorem]], though this theorem is itself quite hard to prove. D.J. Newman observed that the full strength of Ikehara's theorem is not needed for the prime number theorem, and one can get away with a special case that is much easier to prove.