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Newman's proof of the Prime Number Theorem
The method of 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 the critical estimate uses only elementary techniques from a first course in the subject: Cauchy's integral formula and Cauchy's integral theorem and estimates of complex integrals. Here is a brief sketch of this proof.

The first and second Chebyshev function are respectively

\psi(x) = \sum_{k \ge 1}\sum_{p^k\le x}\log p \quad \text{ and     }  \quad \vartheta(x) = \sum_{p\le x} \log p. $$ The second series is obtained by dropping the terms with $$k \ge 2$$ from the first one. PNT is equivalent to either $$\lim _{x \to \infty} \psi(x)/x = 1$$ or $$\lim _{x \to \infty} \vartheta(x)/x = 1$$.

The sums for $$\psi$$ and $$\vartheta$$  are partial sums of the coefficients of the Dirichlet series

- \frac{\zeta '(s)}{\zeta(s)} = \sum_{k \ge 1}\sum_{p^k\le x}\log p \,\, p^{-ks} \quad \text{  and     }  \quad  \quad \Phi(s) = \sum_{p\le x} \log p\,\,  p^{-s}, $$ where $$\zeta$$ is the Riemann zeta function. As with the partial sums, the second series is obtained by dropping the terms with $$k \ge 2$$ from the first one. The Dirichlet series formed by terms with $$k \ge 2$$ is dominated by the Dirichlet series for $$\zeta(2s + \varepsilon)$$  for any positive $$\varepsilon$$, so the logarithmic derivative of $$\zeta$$  and $$\Phi(s)$$  differ by a function holomorphic in $$\Re s > \frac 1 2$$ , and therefore have the same singularities on the line $$\Re  s = 1$$.

Integration by parts gives for $$\Re s > 1$$ ,

\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. $$

All analytic proofs of the Prime Number Theorem use the fact that $$\zeta$$ has no zeroes on the line $$\Re s = 1$$. One further piece of information needed in Newman's proof is that $$\vartheta(x)/x$$ is bounded. This can be easily proved using elementary methods.

Newman's method proves PNT by showing the integral

I = \int_0 ^\infty \left( \frac{\vartheta(e^t)}{e^t} -1 \right) \, dt. $$ converges, and therefore the integrand goes to zero as $$t \to \infty$$. In general, the convergence of the improper integral does not imply that the integrand goes to zero, since it may oscillate, but since $$\vartheta$$  in increasing, it is easy to show in this case.

For $$\Re z > 0$$ let

g_T(z) = \int_0^T \left( \frac{\vartheta(e^t)}{e^t} -1 \right) e^{-zt}\, dt \quad \quad $$ then

\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 $$ which is holomorphic on the line $$\Re z = 0$$. The convergence of the integral $$ I $$ is proved by showing that $$\lim_{T \to \infty} g_T(0) = g(0)$$. This involves change of order of limits since it can be written

\lim_{T \to \infty} \lim_{s \to 0} g_T(z) = \lim_{s \to 0} \lim_{T \to \infty}g_T(z) $$ and therefore classified as a Tauberian theorem.

The difference $$g(0) - g_T(0)$$ is expressed using Cauchy's integral formula and then estimates are applied to the integral. Fix $$R>0$$ and $$\delta >0$$  such that $$g(z)$$  is holomorphic in the region  where $$ |z| \le R \text{ and } \Re z \ge  - \delta$$ and let $$C$$ be its boundary. Since 0 is in the interior, Cauchy's integral formula gives

g(0) - g_T(0) = \frac 1 {2 \pi i }\int_C \left( g(z) - g_T(z) \right ) \frac {dz} z. $$ To get a rough estimate on the integrand, let $$B$$  be an upper bound for $$\vartheta(e^t)/{e^t} -1$$, then for $$\Re z > 0$$

|g(z) - g_T(z)| \le B\int_T ^ \infty e^{-\Re (z) t} \,dt = \frac {B e^{-\Re (z) T}}{\Re z}. $$ This bound is not good enough to prove the result, but Newman had the ingenious idea to introduce the factor

F(z) = e^{zT}\left( 1 + \frac {z^2}{R^2}\right) $$ into the integrand for $$g(0) - g_T(0)$$. Since the Newman factor $$F$$ is entire and $$F(0) =1$$, the left hand side remains unchanged. Now the estimate above for $$|g(z) - g_T(z)|$$ and estimates on $$F$$  combine to give

\left |\frac 1 {2 \pi i }\int_{C_+} \left( g(z) - g_T(z)\right) F(z) \frac {dz} z \right | \le \frac B R. $$ where $$C_+$$ is the semicircle  $$C \cap \left \{ z \, \vert \, \Re z >  0 \right \}$$.

Let $$C_-$$ be the contour $$C \cap \left \{ \Re z \le 0 \right \}$$. The function $$g_T$$ is entire so by Cauchy's integral theorem, the contour $$C_-$$  can be modified to a semicircle of radius $$R$$  in the left half-plane without changing the integral of $$g_T(z)F(z)/2 \pi i z$$, and the same argument as above gives that the absolute value of this integral is $$\le B/R$$. Finally letting $$T \to \infty$$, the integral of $$g(z)F(z)/z$$  over the contour $$C_\delta$$  goes to zero since $$F$$  goes to zero on the contour. Combining the three estimates, get

\limsup_{T \to \infty }|g(0) - g_T(0) | \le \frac {2 B} R. $$ This holds for any $$R$$  so $$\lim_{T \to \infty} g_T(0) = g(0)$$, and PNT follows.