Riemann zeta function
The Riemann zeta function, named after the mathematician Bernhard Riemann, is a mathematical function that plays a crucial role in number theory and analysis. It is defined for complex numbers s with real part greater than 1, and can be extended to a meromorphic function for all complex numbers except s=1, where it has a simple pole.
About
It is defined for complex numbers s with real part greater than 1, and can be extended to a meromorphic function for all complex numbers except s=1, where it has a simple pole. The zeta function is most widely known for its relationship with the distribution of prime numbers, as showcased in Riemann's famous hypothesis. This hypothesis suggests that all non-trivial zeros of the zeta function have a real part equal to 1/2, which has far-reaching implications in number theory. The Riemann zeta function has numerous other important properties and connections to various mathematical fields, such as complex analysis, harmonic series, and functional equations. It has also found applications in physics, particularly in quantum field theory and statistical mechanics. Overall, the Riemann zeta function is a fundamental mathematical function that continues to intrigue and inspire researchers across different disciplines.