Prime number theorem

This proof is a logical sketch of the analytic route, not a self-contained derivation — it needs complex analysis introduced later in the curriculum (see the topic on the Riemann zeta function for the details). The argument has three stages.

Stage 1 (reduce to θ): a routine partial-summation argument shows $\pi(x) \sim x/\log x$ is equivalent to $\theta(x) \sim x$, where $\theta(x)=\sum_{p\le x}\log p$ as in Chebyshev's bound above.

Stage 2 (encode θ via ζ): Riemann's explicit formula expresses $\theta(x)$ (more precisely the closely related $\psi(x)=\sum_{p^k\le x}\log p$) almost exactly in terms of the zeros of $\zeta(s)=\sum_n n^{-s}=\prod_p(1-p^{-s})^{-1}$: the leading term $x$ comes from $\zeta$'s simple pole at $s=1$, and every zero $\rho=\beta+i\gamma$ of $\zeta$ contributes an oscillating error term of size roughly $x^{\beta}$.

Stage 3 (the key nonvanishing fact): $\psi(x)\sim x$ — and hence the theorem — holds if and only if no zero has $\beta=1$, i.e. $\zeta(1+it) \ne 0 \text{ for all } t \in \mathbb{R}$. Hadamard and de la Vallée Poussin proved this nonvanishing independently in 1896, using an elementary trigonometric inequality ($3+4\cos\theta+\cos2\theta\ge0$) applied to $\log|\zeta(\sigma)^3\zeta(\sigma+it)^4\zeta(\sigma+2it)|$ as $\sigma\to1^+$: a zero at $1+it$ would force this expression to $-\infty$, which the inequality forbids. Newman's 1980 proof streamlines Stage 3's consequences into a short Tauberian argument, but the logical backbone — pole at $1$, zeros elsewhere, nonvanishing on the line $\operatorname{Re}(s)=1$ — is exactly as Riemann and Hadamard laid it out.