What is the domain of the Riemann zeta function?

What is the domain of the Riemann zeta function?

The domain of ζ, considered as a complex function, is {s ∈ C | Re(s) > 1}.

Is Riemann zeta function analytic?

The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ > 1. Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

Is Riemann zeta function symmetric?

As far as I learned from the literature, the non-trivial zeros of the zeta function are symmetric about the critical line Re(s) = 1/2, because xi(s) = xi(1-s). Instead the zeros are symmetric about Re(s) = 1/2 AND Im(s) = 0.

Will the Riemann hypothesis be proved?

Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much.

Is the Riemann zeta function finite or infinite?

The Riemann zeta function is defined by (1.61) ζ(s) = 1 + 1 2s + 1 3s + 1 4s + ⋯ = ∞ ∑ k = 1 1 ks. The function is finite for all values of s in the complex plane except for the point s = 1. Euler in 1737 proved a remarkable connection between the zeta function and an infinite product containing the prime numbers:

Is the Riemann zeta function holomorphic to the s-plane?

Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

What is the Riemann hypothesis for the Euler zeta function?

The Riemann hypothesis for the Euler zeta function is a corollary. 1.GeneralizationoftheGammaFunction The Riemann hypothesis is the conjecture made by Riemann that the Euler zeta func- tion has no zeros in a half–plane larger than the half–plane which has no zeros by the convergence of the Euler product.

What is the relation between zeta function and prime numbers?

In 1737, the connection between the zeta function and prime numbers was discovered by Euler, who proved the identity where, by definition, the left hand side is ζ(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products ):