On some new properties of the gamma function and the Riemann zeta function

In this paper, we have exhibited, by utilizing value distribution theory, some new properties of the Gamma function Γ( z ) and the Riemann zeta function ζ( z ). Specifically, we have proved that both of the two functions are prime and the Riemann zeta function, like Γ( z ), does not satisfy any algebraic differential equation with coefficients in ℒ 0 . Moreover, the two functions do not satisfy any functional equation of the form P (Γ, ζ, z ) ≡ 0, where P ( x , y , z ) is a nonconstant polynomial in x , y and z .