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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. The Riemann zeta function is a function of complex argument s that analytically continues the sum of the infinite series sum_{n=1}^{infty}frac{1}{n^s}, quad Re(s)1. It plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. First results about this function were obtained by Leonhard Euler in the eighteenth century. It is named after Bernhard Riemann, who in the memoir "On the Number of Primes Less Than a Given Magnitude", published in 1859, established a relation between its zeros and the distribution of prime numbers.