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.