The Riemann Zeta Function and the Prime Number Theorem

The single function $\zeta (s)=\sum n^{-s}$ ties the analytic machinery of the folder to the distribution of prime numbers. Its Euler product encodes unique factorization, its analytic continuation and functional equation display a hidden symmetry, and the location of its zeros controls the error term in the prime number theorem. This is the showcase application of continuation, products, and contour methods; it builds on gamma-function.md.

Definition and the Euler product

For $\mathrm{Re}s>1$ the series converges absolutely and factors over primes: $$\zeta (s)=\sum _{n=1}^{\infty }\frac{1}{n^s}=\prod _{p\text{prime}}\frac{1}{1-p^{-s}}.$$

The Euler product is analytic unique factorization. Expanding each geometric factor $\frac{1}{1-p^{-s}}={\sum }_k{p}^{-ks}$ and multiplying reproduces every $n^{-s}$ exactly once — because every integer factors into primes in exactly one way. That $\zeta$ has no zeros for $\mathrm{Re}s>1$ (a convergent product of nonzero factors) is already an arithmetic statement: there are infinitely many primes (else the product, and $\zeta$, would be finite at $s=1$, but $\zeta (1)$ is the divergent harmonic series).

Analytic continuation and the functional equation

Multiplying by the Gamma factor and symmetrizing defines the completed zeta function $$\xi (s)={\pi }^{-s/2}\Gamma \left(\frac{s}{2}\right)\zeta (s),$$ which continues to a meromorphic function on $\mathbb{C}$ (simple poles at $s=0,1$) and satisfies the functional equation

Riemann's functional equation. $\xi (s)=\xi (1-s)$, equivalently $$\zeta (s)=2^s{\pi }^{s-1}\sin \left(\frac{\pi s}{2}\right)\Gamma (1-s)\zeta (1-s).$$

It reflects the plane across the line $\mathrm{Re}s=\frac{1}{2}$. One proof runs the integral for $\Gamma (s/2)\zeta (s)$ into the Jacobi theta function and uses its modular transformation $\theta (1/t)=\sqrt{t}\theta (t)$ — the same Poisson-summation symmetry met in evaluating-integrals.md. $\zeta$ continues to a meromorphic function whose only pole is a simple one at $s=1$ with residue $1$.

Zeros and the Riemann hypothesis

The functional equation and the Gamma poles force the structure of the zeros:

• Trivial zeros at $s=-2,-4,-6,\dots$ (from the $\sin (\pi s/2)$ factor).
• Non-trivial zeros all lie in the critical strip $0<\mathrm{Re}s<1$, and are symmetric about the critical line $\mathrm{Re}s=\frac{1}{2}$ (by $\xi (s)=\xi (1-s)=\overline{\xi (\bar{s})}$).

Riemann hypothesis. All non-trivial zeros lie on the line $\mathrm{Re}s=\frac{1}{2}$.

Unproved, and equivalent to the sharpest possible error term in the prime count. The argument principle and Hadamard factorization of $\xi$ (an entire function of order $1$) convert statements about zeros into explicit formulas for prime sums.

The prime number theorem

The link to primes is the logarithmic derivative, whose poles sit at the zeros: $$-\frac{{\zeta }^{\prime }(s)}{\zeta (s)}=\sum _n\frac{\Lambda (n)}{n^s},\Lambda (n)=\{\begin{array}{ll}\log p& n=p^k\\ 0& \text{else}\end{array}$$ (the von Mangoldt function). A contour integral of this against $x^s/s$ picks up the pole at $s=1$ (residue $x$) plus contributions from the zeros. The decisive analytic input is:

Non-vanishing on the edge. $\zeta (s)\ne 0$ for $\mathrm{Re}s=1$.

With no zeros on the line $\mathrm{Re}s=1$, the zero contributions are subdominant and the pole at $s=1$ wins, giving the

Prime number theorem. $\pi (x)\sim \frac{x}{\log x}$, equivalently $\psi (x)={\sum }_{n\le x}\Lambda (n)\sim x$.

The size of the error term is exactly the reach of the zero-free region — and the Riemann hypothesis is the assertion that the error is $O(\sqrt{x}{\log }^{2}x)$. This is the point where complex analysis becomes analytic number theory.

References

• Titchmarsh, The Theory of the Riemann Zeta-Function.
• Stein & Shakarchi, Complex Analysis, Ch. 6–7 (continuation, PNT).
• Davenport, Multiplicative Number Theory, Ch. 7–18.