Editing Prime number theorem (section)

== Newman's proof of the prime number theorem ==

[[Donald J. Newman|D. J. Newman]] gives a quick proof of the prime number theorem (PNT). The proof is "non-elementary" by virtue of relying on complex analysis, but uses only elementary techniques from a first course in the subject: [[Cauchy's integral formula]], [[Cauchy's integral theorem]] and estimates of complex integrals. Here is a brief sketch of this proof.  See <ref name=":0" /> for the complete details.

The proof uses the same preliminaries as in the previous section except instead of the function <math display="inline">\psi</math>, the  [[Chebyshev function]]<math display="inline">
\quad \vartheta(x) = \sum_{p\le x} \log p</math> is used, which is obtained by dropping some of the terms from the series for <math display="inline">\psi</math>.  Similar to the argument in the previous proof based on Tao's lecture, we can show that {{math|''ϑ''&hairsp;&hairsp;(''x'') ≤ ''π''(''x'')log ''x''}}, and {{math|''ϑ''&hairsp;&hairsp;(''x'') ≥ (1 − ''ɛ'')(''π''(''x'') + O(''x''<sup>&hairsp;1 − ''ɛ''</sup>))log ''x''}} for any {{math|0 < ''ɛ'' < 1}}. Thus, the PNT is equivalent to <math>\lim _{x \to \infty} \vartheta(x)/x = 1</math>.
Likewise instead of <math> - \frac{\zeta '(s)}{\zeta(s)} </math> the function <math> \Phi(s) = \sum_{p\le x} \log p\,\,  p^{-s} </math> is used, which is obtained by dropping some terms in the series for <math> - \frac{\zeta '(s)}{\zeta(s)} </math>.  The functions <math> \Phi(s) </math>  and <math> -\zeta'(s)/\zeta(s) </math> differ by a function holomorphic on <math>\Re s = 1</math>.  Since, as was shown in the previous section,  <math>\zeta(s)</math>  has no zeroes on the line <math>\Re s = 1</math>, <math> \Phi(s) - \frac 1{s-1} </math> has no singularities on <math>\Re s = 1</math>.

One further piece of information needed in Newman's proof, and which is the key to the estimates in his simple method, is that <math>\vartheta(x)/x</math>  is bounded. This is proved using an ingenious and easy method due to Chebyshev.

Integration by parts  shows how <math>\vartheta(x)</math> and  <math>\Phi(s)</math> are related.  For <math>\Re s > 1</math>, 
: <math>
\Phi(s) = \int _1^\infty x^{-s} d\vartheta(x)  =  s\int_1^\infty \vartheta(x)x^{-s-1}\,dx = s \int_0^\infty \vartheta(e^t) e^{-st} \, dt.
</math>

Newman's method proves the PNT by showing the integral 
: <math>
I = \int_0 ^\infty \left( \frac{\vartheta(e^t)}{e^t} -1 \right) \, dt.
</math>
converges, and therefore the integrand goes to zero as <math>t \to \infty</math>, which is  the PNT. In general, the convergence of the improper integral does not imply that the integrand goes to zero at infinity, since it may oscillate, but since <math>\vartheta</math>  is increasing, it is easy to show in this case.

To show the convergence of <math> I </math>, for <math>\Re z > 0</math>  let 
: <math> g_T(z) = \int_0^T f(t) e^{-zt}\, dt </math> and <math> g(z) = \int_0^\infty f(t) e^{-zt}\, dt </math> where <math> f(t) = \frac {\vartheta(e^t)}{e^t} -1 </math>
then
: <math> \lim_{T \to \infty} g_T(z) = g(z) =  \frac{\Phi(s)}{s} - \frac 1 {s-1}  \quad \quad  \text{where} \quad  z = s -1 </math>
which is equal to a function holomorphic on the line <math>\Re z = 0</math> .

The convergence of the integral <math> I </math>, and thus the PNT, is proved by showing that <math>\lim_{T \to \infty} g_T(0) = g(0)</math>. This involves change of order of limits since it can be written <math display="inline"> \lim_{T \to \infty} \lim_{z \to 0} g_T(z) = \lim_{z \to 0} \lim_{T \to \infty}g_T(z) </math> and therefore classified as a [[Abelian and Tauberian theorems|Tauberian theorem.]]

The difference <math>g(0) - g_T(0)</math>  is expressed using [[Cauchy's integral formula]] and then shown to be small for <math> T </math> large by estimating the integrand. Fix <math>R>0</math>  and <math>\delta >0</math>  such that <math>g(z)</math>  is holomorphic in the region  where <math> |z| \le  R \text{ and } \Re z \ge  - \delta</math>,  and let <math>C</math>  be the boundary of this region.  Since 0 is in the interior of the region, [[Cauchy's integral formula]] gives 
: <math> g(0) - g_T(0) = \frac 1 {2 \pi i }\int_C \left( g(z) - g_T(z) \right ) \frac {dz} z = \frac 1 {2 \pi i }\int_C \left( g(z) - g_T(z) \right ) F(z)\frac {dz} z </math>
where <math>  F(z) = e^{zT}\left( 1 + \frac {z^2}{R^2}\right) </math> is the factor introduced by Newman, which does not change the integral since <math>F</math> is [[Entire function|entire]] and <math>F(0) = 1</math>.

To estimate the integral, break the contour <math> C </math> into two parts, <math> C = C_+ + C_- </math> where <math>C_+ = C \cap \left \{ z \, \vert \, \Re z >  0 \right \}</math>  and <math>C_- \cap \left \{ \Re z \le 0 \right \}</math>.  Then <math>g(0)- g_T(0) = \int_{C_+}\int_T^\infty H(t,z) dt dz - \int_{C_-}\int_0^T H(t,z) dt dz + \int_{C_-}g(z)F(z)\frac {dz}{2\pi i z}</math>where <math>H(t,z) = f(t)e^{-tz}F(z)/2 \pi i</math>.  Since <math>\vartheta(x)/x</math>, and hence  <math> f(t) </math>, is bounded, let <math>B</math>  be an upper bound for the absolute value of <math>f(t)</math>. This bound together with the estimate  <math> |F| \le 2 \exp(T \Re z)|\Re z|/R </math> for <math> |z| = R </math> gives that the first integral in absolute value is <math>\le B/R</math>.  The integrand over <math>C_-</math> in the second integral is [[Entire function|entire]], so by [[Cauchy's integral theorem]], the contour <math>C_-</math>  can be modified to a semicircle of radius <math>R</math>  in the left half-plane without changing the integral, and the same argument as for the first integral gives the absolute value of the second integral is <math>\le B/R</math>.  Finally, letting <math>T \to \infty</math> , the third integral goes to zero since <math>e^{zT}</math>  and hence <math>F</math> goes to zero on the contour. Combining the two estimates and the limit get
: <math> \limsup_{T \to \infty }|g(0) - g_T(0) | \le \frac {2 B} R. </math>
This holds for any <math>R</math>  so <math>\lim_{T \to \infty} g_T(0) = g(0)</math>, and the PNT follows.