The Riemann zeta function is one of the most profound and extensively studied objects in analytic number theory, with deep connections to the distribution of prime numbers. Initially defined for values of s with real part greater than 1, it can be extended to a meromorphic function on the entire complex plane, except for a simple pole at s = 1.

A central feature of the zeta function is its functional equation, which relates its values at s and 1-s, revealing a deep symmetry. This equation is often written in a symmetric form involving the Gamma function, a fundamental object in complex analysis that generalizes the factorial. The Gamma function plays a key role in the functional equation and links the behavior of the zeta function to complex analytic properties.

Another crucial element in the theory is the Jacobi theta function, whose transformation properties are essential in deriving the functional equation of the zeta function.