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It seems like zeta(n) = (pi)^n / (some number), for even integers n. Can anyone point me to a proof of this?
The zeta function is a mathematical function that is defined as the infinite sum of the reciprocal of the powers of positive integers. It is denoted by the Greek letter "ζ" and was first introduced by the mathematician Leonhard Euler in the 18th century.
The zeta function has many applications in mathematics, particularly in number theory and complex analysis. It is also closely related to the distribution of prime numbers and has connections to other important mathematical concepts such as the Riemann hypothesis and the Basel problem.
The zeta function can be calculated using the infinite series formula or by using the functional equation which relates it to the gamma function. It can also be evaluated using various numerical methods, such as the Euler-Maclaurin summation formula.
The Riemann zeta function is a special case of the zeta function where the variable "s" is a complex number with a real part greater than 1. This function is named after the mathematician Bernhard Riemann and is closely connected to the distribution of prime numbers.
The zeta function has practical applications in many areas, including physics, engineering, and computer science. For example, it is used in the study of quantum mechanics and in the design of efficient data compression algorithms. It also has applications in signal processing, circuit design, and cryptography.