zetafunktsioon

The Zetafunktsioon, often referred to as the Riemann Zeta function, is a mathematical function of a complex variable s, denoted by ζ(s). It is defined for Re(s) > 1 by the infinite series Σ n^(-s), where the sum is taken over all positive integers n. This series converges absolutely for all complex numbers s with a real part greater than 1.

A crucial property of the Zetafunktsioon is its analytic continuation to the entire complex plane, except for a simple pole at s = 1. This continuation allows the function to be defined and studied for values of s where the original series does not converge. The value at the pole is lim s→1 (ζ(s) - 1/(s-1)) = γ, where γ is the Euler-Mascheroni constant.

The Zetafunktsioon plays a pivotal role in number theory, particularly in the distribution of prime numbers. The famous Riemann Hypothesis, one of the most important unsolved problems in mathematics, conjectures that all non-trivial zeros of the Zetafunktsioon lie on the critical line Re(s) = 1/2. The zeros of the Zetafunktsioon are the complex numbers s for which ζ(s) = 0. There are trivial zeros at the negative even integers (-2, -4, -6, ...).

Beyond number theory, the Zetafunktsioon appears in various areas of physics, including quantum mechanics and statistical mechanics, and in fields like signal processing and financial mathematics. Its deep connections to fundamental mathematical structures make it a subject of ongoing research and fascination.