Zeta

Zeta is the sixth letter of the Greek alphabet, represented by the symbol Ζ in uppercase and ζ in lowercase. It is used widely in mathematics, science, and engineering as a symbol for variables, constants, and special functions. The name derives from an earlier Phoenician letter zayin via Greek.

In mathematics, the most famous zeta is the Riemann zeta function, denoted ζ(s). It is defined for complex numbers s with real part greater than 1 by ζ(s) = sum_{n=1}^∞ n^{-s}, and is extended to the complex plane by analytic continuation. The function satisfies an Euler product over primes and encodes information about prime numbers. Its nontrivial zeros lie in the critical strip 0 < Re(s) < 1; the Riemann Hypothesis posits that all such zeros have Re(s) = 1/2. Special values include ζ(2) = π^2/6 and ζ(-1) = -1/12. Related functions include the Hurwitz zeta ζ(s,a) and the Dirichlet eta function, which are connected through continuation.

In physics and chemistry, "zeta" appears in zeta potential, the electrokinetic potential at the boundary of a particle in a liquid, used as an indicator of colloidal stability. Zeta also appears in mathematical physics through zeta function regularization, a method for assigning finite values to certain divergent sums or integrals in quantum field theory.

In astronomy, Greek letters such as zeta designate stars within a constellation in order of apparent brightness (the Bayer designation). Examples include Zeta Tauri and Zeta Ursae Majoris.

Other uses include the zeta distribution in probability (a discrete power-law, also related to Zipf's law) and various concepts named after zeta in engineering and information theory.