The Riemann Zeta Function

Explore the most important function in number theory: from Euler's product to the Riemann Hypothesis

The Most Important Function

The Riemann zeta function ζ(s) = Σ n-s is arguably the most important function in all of mathematics. Euler discovered that it equals an infinite product over primes, revealing a profound connection between the additive structure of integers and the multiplicative structure of primes.

Riemann's 1859 paper showed that the zeros of ζ(s) control the distribution of primes. The Riemann Hypothesis — that all non-trivial zeros have real part 1/2 — is the most famous unsolved problem in mathematics. This page connects directly to Complex Analysis (domain coloring, analytic continuation) and Real Analysis (series convergence).

Demo 1: Zeta Function Domain Coloring

Visualize ζ(s) on the complex plane using domain coloring: hue represents the argument and brightness represents the magnitude. Zeros appear where all colors converge. The dashed line marks Re(s) = 1/2 — the critical line where the Riemann Hypothesis places all non-trivial zeros.

Domain coloring of ζ(s): hue = argument, brightness = magnitude. Dashed line: Re(s) = 1/2 (critical line).

Zeros appear as points where all colors meet. The Riemann Hypothesis asserts all non-trivial zeros lie on the dashed line.

Demo 2: The Euler Product

Euler's stunning discovery: ζ(s) = Πp prime 1/(1 - p-s). Adding one prime at a time to the product, watch it converge to ζ(s). This formula is the bridge between the zeta function and prime numbers.

Primes included
0
Partial product
1.000000
ζ(2.0) ≈
1.644834
error: 6.45e-1
ζ(2.0)

ζ(s) = ∏p prime 1/(1 - p-s)

Demo 3: Zeros on the Critical Line

The Hardy Z-function tracks ζ(1/2 + it) along the critical line. Each sign change corresponds to a zero of ζ. Click the red dots to explore individual zeros. Over 10 trillion zeros have been computed — all on the critical line, consistent with the Riemann Hypothesis.

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The Hardy Z-function: sign changes (red dots) correspond to zeros of ζ(1/2 + it). The Riemann Hypothesis asserts that all non-trivial zeros lie on Re(s) = 1/2.

Demo 4: The Explicit Formula

The deepest result in analytic number theory: each zeta zero contributes an oscillatory correction to the prime counting function π(x). Slide the zero count from 0 to 30 and watch the smooth Li(x) curve develop the jumps and wiggles of π(x). The zeros of zeta literally control where primes appear.

032π(x) (exact)Explicit formula (5 zeros)
γ1 ≈ 14.1γ2 ≈ 21.0γ3 ≈ 25.0γ4 ≈ 30.4γ5 ≈ 32.9

Each zeta zero contributes an oscillatory correction to the prime counting function. As more zeros are added, the approximation better captures the step function π(x).

The Zeta Landscape!

You've explored the crown jewel of analytic number theory:

  • Domain coloring reveals the analytic structure of ζ(s)
  • The Euler product connects ζ directly to prime numbers
  • Non-trivial zeros lie on the critical line (Riemann Hypothesis)
  • The explicit formula reconstructs π(x) from zeta zeros

Next: We turn to algebraic questions — quadratic residues and Gauss's "Golden Theorem" of quadratic reciprocity.