How does the simulation compute zeta(s) inside the critical strip?

The plain series 1/1^s + 1/2^s + 1/3^s + ... only converges for Re(s) greater than 1, so it cannot be summed directly inside the strip. The code instead sums the alternating Dirichlet eta series eta(s) = sum of (-1)^(n-1) / n^s for n from 1 up to the Terms value, which converges for Re(s) greater than 0, then divides by the factor (1 - 2^(1-s)) using complex division to recover zeta(s) = eta(s) / (1 - 2^(1-s)).

Why does the magnitude of zeta on the critical line dip to zero at specific heights?

In the Critical line view the point s = one half plus it climbs straight up the critical line as t increases, and the plot tracks the modulus of zeta at that point. The simulation marks ten known non-trivial zeros at t approximately 14.13, 21.02, 25.01, 30.42, 32.94, 37.59, 40.92, 43.33, 48.01 and 49.77; every time the walking point reaches one of these heights, the modulus genuinely falls to zero because s is a root of the zeta function there.

Riemann Zeta Function — Critical Strip & Zeros

ζ Riemann Zeta

Critical strip & zeros

t = 0.00

|ζ(½+it)| = —

View

Parameters

Terms 60

Zoom (max Im) 30

Speed 1.0

Controls

Stats

t (height)

0.00

|ζ(½+it)|

—

Zeros passed

Status

Ready

First non-trivial zeros

Info & Theory

The Riemann zeta function starts life as the series ζ(s) = Σ 1/nˢ, which only converges for Re(s) > 1. Analytic continuation extends it to the whole complex plane except a pole at s = 1.

Reaching the critical strip

To enter 0 < Re(s) < 1 this simulation uses the Dirichlet eta series η(s) = Σ (−1)ⁿ⁻¹/nˢ (convergent for Re(s) > 0) and the identity ζ(s) = η(s) / (1 − 2¹⁻ˢ).

Domain colouring

Every pixel is a complex number s; its colour encodes ζ(s). Hue is the argument (phase) and lightness is the modulus. Zeros show up as dark points where every hue collides.

The critical line

The vertical white line marks Re(s) = ½. The Riemann hypothesis says every non-trivial zero lies exactly here. The first few sit at heights 14.13, 21.02, 25.01, 30.42, 32.94, ….

Walking the line

Switch to the critical-line view and a point climbs up ½ + it while the plot of |ζ(½+it)| dips to zero at each zero — these zeros control the fine distribution of the prime numbers.

Frequently asked questions

What is the Riemann zeta function?

The Riemann zeta function ζ(s) is defined for Re(s) > 1 by the series 1 + 1/2ˢ + 1/3ˢ + … and extended to the whole complex plane (except s = 1) by analytic continuation. Through its Euler product it encodes deep information about the prime numbers.

What is the critical strip and the critical line?

The critical strip is the region 0 < Re(s) < 1. The critical line is Re(s) = ½, the vertical line down its centre. The Riemann hypothesis states that every non-trivial zero of ζ lies exactly on this line.

What are the non-trivial zeros?

Besides the trivial zeros at s = −2, −4, −6, …, ζ has infinitely many non-trivial zeros inside the critical strip. The first few sit at roughly ½ + 14.13i, ½ + 21.02i, ½ + 25.01i and so on — all so far found on the critical line.

How does this simulation continue ζ into the strip?

The plain series only converges for Re(s) > 1. The simulation uses the Dirichlet eta function η(s) = Σ (−1)ⁿ⁻¹/nˢ, which converges for Re(s) > 0, and the identity ζ(s) = η(s) / (1 − 2¹⁻ˢ) to reach the critical strip.

What does domain colouring show?

Each point s on the plane is coloured by the value ζ(s): the hue encodes the argument (phase) and the lightness encodes the modulus |ζ(s)|. Zeros appear as points where all hues meet and the colour goes dark.

Why does |ζ(½+it)| dip to zero?

As you walk a point up the critical line, the side plot tracks |ζ(½+it)|. Each time t reaches the imaginary part of a zero, the modulus drops to zero, marking that non-trivial zero.

Why does the number of terms matter?

The eta series converges slowly near the critical strip, so more terms give a more accurate value of ζ but cost more computation. The term-count slider trades accuracy for speed.

What is the Riemann hypothesis?

The Riemann hypothesis conjectures that all non-trivial zeros of ζ have real part exactly ½. It is one of the seven Millennium Prize Problems and controls how regularly the primes are distributed.

How are the zeros tied to the prime numbers?

Riemann's explicit formula expresses the count of primes up to a number as a main term plus oscillating corrections, one for each zero. The location of the zeros therefore directly controls the error in the prime-counting estimate.

Is this an exact computation of ζ?

It is a faithful numerical approximation, not exact arithmetic. With enough terms the colouring and the |ζ| plot match the true zeta function closely in the region shown, but very high up the critical line you would need more advanced summation.

About this simulation

Last updated: 9 July 2026

This visualiser plots the Riemann zeta function ζ(s) over the complex plane by domain colouring: every pixel is a complex number s = re + i·im, and its hue encodes the argument (phase) of ζ(s) while its lightness encodes the modulus |ζ(s)|. Because the defining series 1 + 1/2ⁿ + 1/3ⁿ + … only converges for Re(s) > 1, the simulation instead sums the Dirichlet eta series η(s) = Σ(−1)ⁿ⁻¹/nⁿ, which converges for Re(s) > 0, and recovers ζ via the identity ζ(s) = η(s) / (1 − 2¹⁻ⁿ). This lets it reach into the critical strip 0 < Re(s) < 1 and reveal the non-trivial zeros that sit — as far as anyone has ever checked — exactly on the critical line Re(s) = ½.

🔬 What it shows

A domain-coloured map of ζ(s): the vertical white line marks the critical line Re(s)=½, the two dashed lines mark the edges of the critical strip (Re(s)=0 and Re(s)=1), and white dots mark the first ten known non-trivial zeros, all lying on that single line. Switching to the "Critical line" view instead plots |ζ(½+it)| as a curve against height t, dipping to zero at each zero.

🎮 How to use it

The Terms slider (10–200) sets how many terms of the eta series are summed — more terms trade speed for accuracy. Zoom (max Im) (10–50) sets how high up the imaginary axis the view extends. Speed controls how fast the yellow point climbs the critical line. Walk line animates the climb, Step advances it once, and Reset returns to t=0; the View buttons switch between the complex-plane map and the critical-line plot.

💡 Did you know?

Mathematicians have verified over 10 trillion non-trivial zeros of ζ(s) by computer, and every single one has landed exactly on the critical line Re(s)=½. Proving this holds for all of the infinitely many zeros — the Riemann hypothesis — is one of the seven Clay Millennium Prize Problems, worth one million US dollars, and remains unsolved since it was first proposed in 1859.

Frequently asked questions

How does the simulation compute ζ(s) inside the critical strip?

The plain series 1/1ⁿ + 1/2ⁿ + 1/3ⁿ + … only converges for Re(s) > 1, so it cannot be summed directly inside the strip. The code instead sums the alternating Dirichlet eta series η(s) = Σ(−1)ⁿ⁻¹ n⁻ⁿ for n from 1 up to the Terms value, which converges for Re(s) > 0, then divides by the factor (1 − 2¹⁻ⁿ) using complex division to recover ζ(s) = η(s) / (1 − 2¹⁻ⁿ).

What do the hue and lightness of each pixel mean?

Each pixel corresponds to a complex point s, and its colour is computed directly from ζ(s). The hue is set by the argument (angle) of ζ(s) in the complex plane, mapped across the full 360° colour wheel, while the lightness is a compressed function of the modulus |ζ(s)|: as |ζ(s)| shrinks toward zero the pixel darkens toward black, which is why zeros of ζ appear as small dark points where every hue meets.

What are the critical strip and the critical line?

The critical strip is the vertical band 0 < Re(s) < 1, shown in the simulation between two dashed lines. The critical line is the line Re(s)=½ running down its centre, drawn in solid white. The unproven Riemann hypothesis states that every non-trivial zero of ζ(s) lies exactly on this critical line rather than merely somewhere inside the strip.

Why does |ζ(½+it)| dip to zero at specific heights?

In the "Critical line" view the point s=½+it climbs straight up the critical line as t increases, and the plot tracks the modulus |ζ(½+it)|. The simulation marks ten known non-trivial zeros at t ≈ 14.13, 21.02, 25.01, 30.42, 32.94, 37.59, 40.92, 43.33, 48.01 and 49.77; every time the walking point reaches one of these heights, |ζ| genuinely falls to zero because s is a root of the zeta function there.

Why does changing the Terms slider change the picture?

The Dirichlet eta series is an infinite sum, and the simulation can only add up a finite number of terms (10 to 200, chosen by the slider) before stopping. With few terms the partial sum is a rougher approximation to the true ζ(s), which can slightly blur colour boundaries and zero positions; with more terms the sum converges closer to the exact value at the cost of more arithmetic per pixel, which is why higher Terms settings render more slowly.

How are the zeros of ζ(s) connected to prime numbers?

Through its Euler product, ζ(s) equals an infinite product over all primes p of 1/(1−p⁻ⁿ), which is why its zeros are not arbitrary: Riemann's explicit formula expresses the exact count of primes up to any number as a smooth main term plus a correction built from oscillating terms, one per non-trivial zero. The precise locations of the zeros shown in this simulation therefore directly control how evenly or unevenly the primes are distributed.