zetafunkcióval

The Riemann zeta function, denoted by the Greek letter zeta (ζ), is a function of a complex variable s. In its simplest form, for complex numbers s with a real part greater than 1, it is defined by the infinite series: ζ(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ...

This series converges absolutely for Re(s) > 1. However, the Riemann zeta function can be extended to all complex numbers except for s = 1, where it has a simple pole. This extension is typically achieved through analytic continuation.

One of the most significant properties of the zeta function is its connection to prime numbers. The Euler product formula states that for Re(s) > 1, ζ(s) is equal to the infinite product over all prime numbers p: ζ(s) = (1 / (1 - 1/p^s)). This formula directly links the zeta function to the distribution of prime numbers.

The Riemann zeta function is famously associated with the Riemann Hypothesis, a conjecture that states all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. The truth of this hypothesis has profound implications for number theory, particularly concerning the distribution of prime numbers. Despite extensive research and computational verification for a vast number of zeros, the Riemann Hypothesis remains unproven.