The Riemann zeta function is a mathematical function that has deep connections with number theory and analysis. It was introduced by the renowned mathematician Bernhard Riemann in 1859, as part of his study of prime numbers and the distribution of their zeros.
The Riemann zeta function is defined for complex numbers s with a real part greater than 1 and is represented by the Greek letter ζ(s). It is typically written as an infinite series, known as the Euler product formula:
ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + 4^(-s) + ...
This series converges for real numbers s greater than 1 and defines the function there. However, the definition can be extended analytically to the entire complex plane, except for the point s = 1, where it has a simple pole.
The Riemann zeta function plays a vital role in number theory, as it encodes the behavior of prime numbers. One of the most significant insights made by Riemann was the connection between the zeros of the zeta function and the distribution of primes. The Riemann Hypothesis, which remains an unsolved problem in mathematics, suggests that all the non-trivial zeros of the zeta function lie on a specific critical line with real part 1/2.
The Riemann zeta function has wide-ranging applications in various areas of mathematics, including complex analysis, harmonic analysis, and the study of the distribution of prime numbers. Its in-depth understanding continues to be an active research topic for mathematicians.
