The prime number theorem identifies the first-order density of primes. Its analytic proof converts primes into the logarithmic derivative of the zeta function and then extracts asymptotics from the behavior at \(s=1\).
Equivalent counting functions
Definition. \[ \pi(x)=\#\{p\le x\},\qquad \vartheta(x)=\sum_{p\le x}\log p,\qquad \psi(x)=\sum_{n\le x}\Lambda(n). \]
Prime number theorem. \[ \pi(x)\sim\frac{x}{\log x}, \qquad\text{equivalently}\qquad \vartheta(x)\sim x \quad\text{or}\quad \psi(x)\sim x. \]
Equivalence. Prime powers with exponent at least \(2\) contribute \[ 0\le\psi(x)-\vartheta(x) \le\sum_{k\ge2}\vartheta(x^{1/k})=O(\sqrt x\log^2x), \] so \(\psi\sim x\) iff \(\vartheta\sim x\). Partial summation gives \[ \pi(x)=\frac{\vartheta(x)}{\log x} +\int_2^x\frac{\vartheta(t)}{t\log^2t}\,\mathrm{d}t. \] If \(\vartheta(x)\sim x\), the integral is lower order, proving \(\pi(x)\sim x/\log x\); the converse follows from the analogous summation identity for \(\vartheta\).
The analytic mechanism
For \(\operatorname{Re}s>1\), \[ -\frac{\zeta'(s)}{\zeta(s)} =\int_{1^-}^{\infty}x^{-s}\,\mathrm{d}\psi(x). \] The simple pole \[ \zeta(s)=\frac1{s-1}+h(s) \] suggests that the transform behaves like \(1/(s-1)\), the transform of the measure \(\mathrm{d}x\). To justify the inverse implication one needs that \(\zeta(s)\ne0\) on \(\operatorname{Re}s=1\).
Theorem (Hadamard-de la Vallee Poussin). \[ \zeta(1+it)\ne0\qquad(t\in\mathbb R,\ t\ne0). \]
Key argument. For \(\sigma>1\), \[ 3+4\cos\theta+\cos2\theta=2(1+\cos\theta)^2\ge0. \] Using the Euler product logarithm yields \[ \zeta(\sigma)^3 |\zeta(\sigma+it)|^4 |\zeta(\sigma+2it)|\ge1. \] If \(\zeta(1+it)=0\) of order \(m\), then as \(\sigma\downarrow1\), the simple pole of \(\zeta(\sigma)^3\) contributes order \(-3\), while the fourth power contributes at least \(4m\), forcing the left side to zero, a contradiction.
Tauberian conclusion
Wiener-Ikehara theorem. If \(A(x)\) is nondecreasing and \[ \int_{1^-}^{\infty}x^{-s}\,\mathrm{d}A(x)-\frac{a}{s-1} \] extends continuously to \(\operatorname{Re}s\ge1\), then \(A(x)\sim ax\).
Apply the theorem with \(A=\psi\) and \(-\zeta'/\zeta\). The pole at \(1\) has residue \(1\), and the zero-free line supplies the boundary regularity, so \(\psi(x)\sim x\).
Error terms and the Riemann hypothesis
Moving contour integrals leftward makes zeros of \(\zeta\) contribute oscillatory terms. The Riemann hypothesis is equivalent to near-square-root error estimates such as \[ \psi(x)=x+O(x^{1/2}\log^2x). \] Thus the main term comes from the pole at \(1\), while the precision of prime counting is governed by the location of nontrivial zeros.