Riemann Zeta Function — Mathematics Knowledge System

Overview

The Riemann zeta function ζ(s) is one of the most important functions in mathematics, connecting number theory, complex analysis, and physics. Initially defined for Re(s) > 1 by the Dirichlet series Σ n^{-s}, it extends to a meromorphic function on ℂ with a single simple pole at s = 1. Its zeros govern the distribution of prime numbers via the explicit formula for π(x). The Riemann Hypothesis — that all non-trivial zeros lie on the critical line Re(s) = 1/2 — is one of the Millennium Prize Problems and remains open.

Intuition

The zeta function encodes information about all integers simultaneously through the series 1 + 1/2^s + 1/3^s + … Its remarkable connection to primes comes from the Euler product: because every integer factors uniquely into primes, the sum over all integers equals a product over all primes. Zeros of ζ are 'resonances' in prime distribution: each zero ρ contributes an oscillating term x^ρ to the prime counting formula. If all zeros lie on Re(s) = 1/2, the oscillations are as small as possible — this is what the Riemann Hypothesis asserts.

Formal Definition

Definition

For Re(s) > 1, the Riemann zeta function is defined by the absolutely convergent Dirichlet series. It admits analytic continuation to ℂ\{1} and satisfies a functional equation.

\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p\text{ prime}} \frac{1}{1 - p^{-s}}, \quad \mathrm{Re}(s) > 1

Dirichlet series and Euler product

\xi(s) = \tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)

Completed zeta function (entire)

\xi(s) = \xi(1-s)

Functional equation

\zeta(s) = \frac{\pi^{s-1/2}\,\Gamma((1-s)/2)}{\Gamma(s/2)}\,\zeta(1-s)

Functional equation in terms of ζ

\psi(x) = x - \sum_{\rho:\,\zeta(\rho)=0} \frac{x^\rho}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \tfrac{1}{2}\ln(1-x^{-2})

Riemann's explicit formula for the Chebyshev psi function

Notation

Notation	Meaning
$\zeta(s)$	Riemann zeta function
$\rho = \beta + i\gamma$	Non-trivial zero of ζ (Riemann Hypothesis: β = 1/2 for all ρ)
$s = \sigma + it$	Complex variable; σ = Re(s), t = Im(s)
$\Gamma(s)$	Euler gamma function (appears in the functional equation)
$\xi(s)$	Completed (entire) zeta function

Properties

Trivial zeros

\zeta(-2n) = 0 \text{ for } n = 1, 2, 3, \ldots

Non-trivial zeros in the critical strip

\text{All non-trivial zeros lie in } 0 < \mathrm{Re}(s) < 1 \text{ (critical strip)}

Special values

\zeta(2) = \frac{\pi^2}{6},\quad \zeta(4) = \frac{\pi^4}{90},\quad \zeta(2n) = \frac{(-1)^{n+1}(2\pi)^{2n}B_{2n}}{2(2n)!}

Simple pole at s = 1

\zeta(s) = \frac{1}{s-1} + \gamma + O(s-1) \quad (s \to 1)

Worked Examples

1
Consider sin(πz)/πz = Π_{n=1}^∞ (1 - z²/n²). Taking the logarithmic derivative and expanding in power series:
$\frac{\sin(\pi z)}{\pi z} = \prod_{n=1}^\infty \left(1 - \frac{z^2}{n^2}\right)$
2
Expand both sides as power series in z². The coefficient of z² on the left is -π²/6; on the right it is -Σ 1/n².
$\frac{\sin(\pi z)}{\pi z} = 1 - \frac{\pi^2 z^2}{6} + \cdots = 1 - \zeta(2)z^2 + \cdots$
3
Comparing coefficients gives ζ(2) = π²/6.
$\zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$

✓ Answer

ζ(2) = π²/6, first proved by Euler in 1734 ('Basel problem').

Practice Problems

Hardfree response

Use the Euler product to show that 1/ζ(s) = Σ μ(n)/n^s where μ is the Möbius function.

Hardfree response

State the Riemann Hypothesis precisely and explain its consequences for the error term in the Prime Number Theorem.

Common Mistakes

Common Mistake

Thinking ζ(1) is finite or that the series Σ 1/n converges

The harmonic series Σ 1/n diverges, so ζ(1) is not defined as a convergent series. The analytic continuation gives a simple pole at s=1 with residue 1, not a finite value.

Common Mistake

Confusing the trivial zeros with the non-trivial zeros

Trivial zeros are at s = -2, -4, -6, … (negative even integers), arising from the gamma function factor. Non-trivial zeros lie in the critical strip 0 < Re(s) < 1 and are the ones relevant to the Riemann Hypothesis and prime distribution.

Quiz

Historical Background

Euler studied the series Σ 1/n^s for real s and proved the product formula ζ(s) = Π (1-p^{-s})^{-1} in 1737, connecting it to primes. Riemann's 1859 paper 'Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse' was a landmark: it extended ζ to the complex plane, proved the functional equation, and related the zeros to prime distribution. The Riemann Hypothesis was stated there and has resisted proof for over 165 years.

1737
Euler proves the product formula ζ(s) = Π(1-p^{-s})^{-1} for real s > 1
Leonhard Euler
1859
Riemann extends ζ to ℂ, proves the functional equation, states the Riemann Hypothesis
Bernhard Riemann
1896
Hadamard and de la Vallée Poussin show ζ(1+it) ≠ 0, proving the PNT
Jacques Hadamard, Charles-Jean de la Vallée Poussin
1914
Hardy proves infinitely many zeros lie on Re(s) = 1/2
G.H. Hardy
2000
The Riemann Hypothesis listed as a Millennium Prize Problem
Clay Mathematics Institute

Summary

ζ(s) = Σ n^{-s} = Π(1-p^{-s})^{-1} for Re(s) > 1; extends meromorphically to ℂ with a simple pole at s=1.
The functional equation ξ(s) = ξ(1-s) relates values at s and 1-s.
Special values: ζ(2n) = (-1)^{n+1}(2π)^{2n} B_{2n} / (2(2n)!); ζ(2) = π²/6.
Non-trivial zeros govern prime distribution via Riemann's explicit formula.
The Riemann Hypothesis (Re(ρ) = 1/2 for all non-trivial zeros) remains one of the most important open problems in mathematics.

References

BookRiemann, B. — Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie (1859)
BookTitchmarsh, E.C. — The Theory of the Riemann Zeta-Function, 2nd ed. rev. Heath-Brown (1986), Oxford University Press
BookApostol, T.M. — Introduction to Analytic Number Theory (1976), Springer, Chapter 12
WebsiteWikipedia — Riemann zeta function
WebsiteMathWorld — Riemann Zeta Function
WebsiteNIST DLMF — Chapter 25: Zeta and Related Functions