Is the zeta function holomorphic?

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

Is the Riemann zeta function continuous?

The Riemann zeta function is the infinite sum of terms 1/ns, n ≥ 1. For each n, the 1/ns is a continuous function of s, i.e. 1 ns = 1 ns0 , for all s0 ∈ C, and is differentiable, i.e.

What does the zeta function do?

Riemann zeta function, function useful in number theory for investigating properties of prime numbers. Written as ζ(x), it was originally defined as the infinite series ζ(x) = 1 + 2−x + 3−x + 4−x + ⋯. When x = 1, this series is called the harmonic series, which increases without bound—i.e., its sum is infinite.

Why is the zeta function important?

The zeta function ζ(s) today is the oldest and most important tool to study the distribution of prime numbers and is the simplest example of a whole class of similar functions, equally important for understanding the deepest problems of number theory.

What is the domain of the zeta function?

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

What does the Riemann zeta function tell us?

Why is Riemann zeta function important?

This is of central importance in mathematics because the Riemann zeta function encodes information about the prime numbers — the atoms of arithmetic. The nontrivial zeros play a central role in an exact formula, first written down by Riemann, for the number of primes in a given range (say, between one and 10 billion).

Has anyone proved the Riemann hypothesis?

Reimann proved this property for the first few primes, and over the past century it has been computationally shown to work for many large numbers of primes, but it remains to be formally and indisputably proved out to infinity.

What does Zeta mean in math?

zeta function, in number theory, an infinite series given by. where z and w are complex numbers and the real part of z is greater than zero.