Zeta Function
The Riemann zeta function is a mathematical function of a complex variable, conventionally denoted ζ(s). It is defined initially for complex numbers with real part greater than 1 through an infinite series, and then extended analytically to nearly all other complex numbers. The function has deep significance in number theory due to its intimate connection with the distribution of prime numbers along the integer line.
Historical Context and Definition
The zeta function was first studied systematically by Leonhard Euler in the 18th century, who established its connection to prime numbers through the Euler product formula. The function was later extended to the complex plane by Bernhard Riemann in 1859, whose work revealed its non-trivial zeros to be of central importance. These zeros—the points where ζ(s) equals zero for certain complex values—hold the key to understanding how primes are distributed.
The Riemann Hypothesis
The Riemann Hypothesis, one of mathematics’ most famous unsolved problems, conjectures that all non-trivial zeros of the zeta function lie on a specific vertical line in the complex plane (where the real part equals 1/2). Despite extensive computational verification and theoretical work over more than 160 years, this conjecture remains unproven. Its resolution would have profound implications for understanding prime number gaps and distribution patterns, with applications extending to cryptography and other fields.