Explore the most important function in number theory: from Euler's product to the Riemann Hypothesis
One function, ζ(s), ties the whole numbers to the primes, and its zeros decide where the primes fall. Where those zeros lie is the Riemann Hypothesis.
Colour shows the angle of ζ(s) and brightness its size. Find the zeros where all colours meet, and check them against the dashed line Re(s) = 1/2.
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.
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).
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.
Add primes to the product one at a time and watch it close in on ζ(s).
ζ(s) = ∏p prime 1/(1 - p-s)
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.
Follow the curve along the critical line: each sign change is a zero of ζ. Click the red dots to inspect individual zeros.
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.
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.
Slide the zero count from 0 to 30 and watch the smooth Li(x) curve grow the jumps and wiggles of π(x).
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 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.
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