# Wilson's theorem and the Zeta function

$$\zeta(s)$$ has a direct link to the prime numbers (via the infinite Euler product and the non-trivial zeros in the explicit $$\pi(x)$$ formula). Wilson's theorem offers a (proven but very inefficient) formula to generate prime numbers.

Why does the Gamma-function complete the Riemann Zeta function?

I wondered whether there could be a connection between the two and I tried the following:

1) Express Wilson's theorem in terms of the $$\Gamma(s)$$ (instead of the factorial) and the $$cos$$ (instead of the mod)

$$P(s) = \cos \left( \frac{\pi}{2} \frac {\Gamma \left( s \right) +1 }{s} \right)$$

This function has unique integer zeros only when $$s$$ is a prime number.

2) Express $$\Gamma(s)$$ in terms of $$\zeta(s)$$ by e.g. using a Mellin transform:

$$\zeta(s) \Gamma(s) = \int_0^\infty \frac{x^{s-1}}{e^x-1} dx$$

$$P(s) = \cos \left[ \frac{\pi}{2} \left(\frac {1}{s \zeta(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1} dx + \frac{1}{s} \right) \right]$$

3) Link the $$\zeta(s)$$ back to the (encoded) prime numbers:

$$\zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} \right)}{2(s-1)\Gamma(1+\frac{s}{2})}$$

to make the link to the non trivial zeros ($$\rho$$) or simply use the infinite Euler product of prime numbers.

This is just playing with formulae, I know, but wondered if anybody knows whether a stronger link could (or does) exist?

• Nope. I don't think there's any hope of making a connection like this. The manner in which $\zeta(s)$ links with primes is very different from the mechanism driving Wilson's theorem. – David Hansen Aug 6 '11 at 20:57
• I guess replacing the faculty by a Gamma function is novel, just don't tell my Dean. – Charlie Frohman Aug 7 '11 at 0:16
• @Agno: gamma(1+x) = factorial(x) =/= "faculty" ;-) – Gottfried Helms Aug 7 '11 at 10:00
• I know. I know. This was a stupid language mistake. In Dutch the word faculty actually has this double connotation. I obviously did mean factorial and the last thing I want to do was to upset some Deans :-) – Agno Aug 7 '11 at 13:02
• I enjoyed the faculty/factorial wordplay, but I thought after seven years it would be OK to edit it (and a couple of small typos). – Gerry Myerson Nov 25 '18 at 22:01