Proof Outline of Prime Number Theorem

We sketch the analytic proof of the Prime Number Theorem, following Hadamard and de la Vallée Poussin.

TheoremPrime Number Theorem (Restated)

\psi(x) \sim x \quad \text{as } x \to \infty

where $\psi(x) = \sum_{n \leq x} \Lambda(n)$ .

ProofProof Sketch via Wiener-Ikehara

Step 1: Connect to Zeta Function

Taking logarithmic derivative of the Euler product:

-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}

By Perron's formula or Mellin inversion:

\psi(x) = \sum_{n \leq x} \Lambda(n) \leftrightarrow -\frac{\zeta'(s)}{\zeta(s)}

Step 2: Establish Zero-Free Region

The key analytic input: prove $\zeta(1 + it) \neq 0$ for all $t \neq 0$ .

Proof idea: Use the inequality

3 + 4\cos(\theta) + \cos(2\theta) = 2(1 + \cos(\theta))^2 \geq 0

Applied to $\zeta(\sigma)^3 |\zeta(\sigma+it)|^4 |\zeta(\sigma+2it)| \geq 1$ , this forces $\zeta(1+it) \neq 0$ .

Step 3: Analytic Continuation

Since $\zeta(s) \neq 0$ on $\Re(s) = 1$ , the function:

F(s) = -\frac{\zeta'(s)}{\zeta(s)}

extends continuously to $\Re(s) \geq 1$ except for a simple pole at $s=1$ with residue $1$ .

Step 4: Apply Wiener-Ikehara

The Wiener-Ikehara theorem (a Tauberian theorem) states: if $F(s)$ has the properties in Step 3, then:

\sum_{n \leq x} \Lambda(n) \sim x

This is precisely PNT.

Step 5: Deduce $\pi(x) \sim x/\log x$

From $\psi(x) \sim x$ , partial summation gives:

\pi(x) \log x = \psi(x) + \int_2^x \frac{\psi(t)}{t} dt = x + o(x) + \int_2^x \frac{t + o(t)}{t} dt \sim x

Therefore $\pi(x) \sim x/\log x$ .

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Remark

The genius of the analytic proof is reducing PNT to:

Algebraic fact: Euler product gives $-\zeta'(s)/\zeta(s) = \sum \Lambda(n)/n^s$
Analytic fact: $\zeta(1+it) \neq 0$ (the zero-free region)
General principle: Wiener-Ikehara Tauberian theorem

Each step is clean and conceptual, unlike elementary proofs which involve intricate combinatorics.

ExampleError Terms

Better zero-free regions yield better error terms:

Classical: $\Re(s) \geq 1 - c/\log(|t|+2)$ gives $\psi(x) = x + O(x e^{-c'\sqrt{\log x}})$
RH: $\Re(s) = 1/2$ for all zeros gives $\psi(x) = x + O(\sqrt{x} \log^2 x)$ (optimal)