Euler: zeta of n is the sum between 1 and inf. of n ^-s.
He showed that this is the same as the product of all the primes (p) for: (1-p^-s)^-1
The Riemann zeta function is the Euler zeta, where s is a complex number: a +it, where a is greater than 1.
A more complex version of this expression can extend to real values of s less than 1, giving values of 0 for all negative even values of the real part of s (these are called the trivial zeros).
There are no 0 values for the function with real part of s greater than 1 and Riemann hypothesized that all non trivial 0 values have real part of s = 1/2.
Some notes about the Riemann hypothesis:
The zeta function is a sum of harmonics, 1/1^s + 1/2^s… and of harmonics of harmonics etc. as s increases.
Riemann looked at the zeta function, 1/1^s + 1/2^s if you fed in complex numbers for s and retrieved complex numbers out. This requires 4 variables, and 4 dimensions to plot.
The value of the function is 0 for negative even integers, -2, -4 etc. (this are trivial zeros), but all the other zeros that have been found, the non-trivial ones, have a real part = 1/2 - but nobody has proved it.
The set of complex numbers is all possible numbers, Gauss proved this in his doctoral thesis. Real numbers are complex numbers that lie on the real axis of the complex plane.
All prime numbers sit on the real axis.
Non prime numbers that sit on the real axis can be considered to be a superimposition of multiple numbers. i.e. 12 is a superimposition of the states 1 x 12, 2 x 6 and 3 x 4.
This is analogous to the superimposition of quantum states.
Prime numbers are like single states that cannot be created by scaling, adding or multiplying a non prime.
The calculations in (Odlyzko 1987) show that the zeros of the zeta function behave very much like the eigenvalues of a random Hermitian matrix, suggesting that they are the eigenvalues of some self-adjoint operator.