Zetafunktsioonile

Zetafunktsioonile, also known as the Riemann zeta function, is a fundamental concept in mathematics, named after the German mathematician Bernhard Riemann. It is defined for complex numbers with a real part greater than 1 as the infinite series:

ζ(s) = ∑ from n=1 to ∞ of 1/n^s

where s is a complex number. The zeta function plays a crucial role in analytic number theory, particularly in the distribution of prime numbers. It has a meromorphic continuation to the entire complex plane, with a simple pole at s = 1.

The Riemann zeta function satisfies several important functional equations. One of the most famous is:

ζ(s) = 2^s * π^(s-1) * sin(πs/2) * Γ(1-s) * ζ(1-s)

where Γ is the Gamma function. This equation relates the values of the zeta function on the line Re(s) = 1/2 to the values on the line Re(s) = 1.

The Riemann hypothesis is a conjecture that all non-trivial zeros of the zeta function have a real part equal to 1/2. This hypothesis has profound implications in number theory and remains one of the most famous unsolved problems in mathematics. The zeta function also has applications in physics, particularly in the study of quantum field theory and string theory.