The function \zeta(n) , known as the Riemann zeta function, is a mathematical function that plays a crucial role in number theory. It is defined for complex numbers and is particularly important for understanding the distribution of prime numbers. For positive integers n , \zeta(n) is the sum of the reciprocals of the n -th powers of the natural numbers, expressed as \zeta(n) = 1^-n + 2^-n + 3^-n + \ldots . The Riemann zeta function is named after the mathematician Bernhard Riemann, who studied its properties in the 19th century. It has deep connections to various areas of mathematics, including analytic number theory and complex analysis. The function is also linked to the famous Riemann Hypothesis, which conjectures about the distribution of its non-trivial zeros.