What is the Euler Product?
The Euler Product is the identity:
ζ(s) = Πp prime (1 − p−s)−1
It expresses the zeta function as an infinite multiplication of prime-based filters. Each prime contributes a “twist” in the complex plane.
ζ(s) = Πp prime (1 − p−s)−1
It expresses the zeta function as an infinite multiplication of prime-based filters. Each prime contributes a “twist” in the complex plane.
What the Animation Represents
• Each concentric circle is the magnitude of the term p−s
• Each rotating dot represents the phase e−it ln p
• The speed decreases with the size of the prime
• The blue vector is the partial product ζₚ(s)
• As primes rotate, ζₚ(s) fluctuates — this mimics the chaotic nature of ζ(s)
Why Small Primes Matter Most
Small primes (2, 3, 5) produce the largest distortions
in ζ(s).
Larger primes refine the structure but do not dominate it.
This is why early partial products already approximate the shape of ζ(s).
Connection to the Riemann Hypothesis
The Euler product converges only for Re(s) > 1, but its structure
influences behavior throughout the critical strip where Re(s)=1/2.
The oscillations in the animation symbolically reflect the complex interplay behind the nontrivial zeros of ζ(s).
The oscillations in the animation symbolically reflect the complex interplay behind the nontrivial zeros of ζ(s).
Why the Vectors Rotate
Because:
p−s = p−σ · e−it ln p
As t changes, each prime factor traces perfect circles in the complex plane. The animation shows this rotation, with larger primes rotating slower since ln(p) grows.
p−s = p−σ · e−it ln p
As t changes, each prime factor traces perfect circles in the complex plane. The animation shows this rotation, with larger primes rotating slower since ln(p) grows.