zetafunktsioone

The zetafunktsioone, more commonly known as the Riemann zeta function, is a complex-valued function of a complex variable. It is defined by the Dirichlet series $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for all complex numbers $s$ with a real part greater than 1. This series converges absolutely and uniformly in this region.

The function can be extended to the entire complex plane, except for a simple pole at $s=1$, through analytic continuation. This extended function is what is usually referred to as the Riemann zeta function. The value at the pole is $\lim_{s \to 1} (s-1)\zeta(s) = 1$.

The Riemann zeta function plays a crucial role in number theory, particularly in the study of the distribution of prime numbers. The famous Riemann hypothesis, which remains unproven, states that all non-trivial zeros of the zeta function lie on the critical line with real part equal to 1/2.

The zetafunktsioone has connections to various other mathematical fields, including analysis, probability theory, and quantum mechanics. Its values at positive even integers are related to powers of $\pi$, for instance, $\zeta(2) = \frac{\pi^2}{6}$ and $\zeta(4) = \frac{\pi^4}{90}$. The values at positive odd integers are not known to have such simple closed-form expressions.