Prime Secret: ζ(s)

Riemann intuitively found the Zeta Function ζ(s), but couldn’t prove it. Computer ‘tested’ it correct up to billion numbers.

$latex zeta(s)=1+frac{1}{2^{s}}+frac{1}{3^{s}}+frac{1}{4^{s}}+dots$

Or equivalently (see note *)

$latex frac {1}{zeta(s)} =(1-frac{1}{2^{s}})(1-frac{1}{3^{s}})(1-frac{1}{5^{s}})(1-frac{1}{p^{s}})dots$

ζ(1) = Harmonic series (Pythagorean music notes) -> diverge to infinity
(See note #)

ζ(2) = Π²/6 [Euler]

ζ(3) = not Rational number.

1. The Riemann Hypothesis:
All non-trivial zeros of the zeta function have real part one-half.

ie ζ(s)= 0 where s= ½ + bi

Trivial zeroes are s= {- even Z}:
s(-2) = 0 =s(-4) =s(-6) =s(-8)…

You might ask why Re(s)=1/2 has to do with Prime number ?

There is another Prime Number Theorem (PNT) conjectured by Gauss and proved by Hadamard and Poussin:

π(Ν) ~ N / log N
ε = π(Ν) – N / log N
The error ε hides in the Riemann Zeta Function’s non-trivial zeroes, which all lie on the Critical…