Zetafunktsiooni

The Zetafunktsioon, also known as the Riemann zeta function, is a mathematical function of a complex variable that plays a crucial role in number theory and has connections to various other areas of mathematics and physics. It is typically denoted by the Greek letter zeta, ζ.

The most common definition of the Riemann zeta function for a complex number s with a real part greater than 1 is given by the infinite series: ζ(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ... This series converges for Re(s) > 1.

Through a process called analytic continuation, the zeta function can be extended to be defined for all complex numbers except for a single pole at s = 1. This extended function is the one most commonly referred to when discussing the Riemann zeta function.

One of the most famous and significant aspects of the Zetafunktsioon is the Riemann Hypothesis. This conjecture states that all non-trivial zeros of the Riemann zeta function lie on the critical line, where the real part of s is exactly 1/2. Proving or disproving this hypothesis would have profound implications for the distribution of prime numbers.

The Zetafunktsioon is deeply connected to prime numbers through the Euler product formula, which expresses the zeta function as an infinite product over all prime numbers: ζ(s) = (1 - 1/2^s)^-1 * (1 - 1/3^s)^-1 * (1 - 1/5^s)^-1 * ... This relationship highlights its fundamental importance in understanding the arithmetic properties of integers.