Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by x=sum_{n=2}^infty q_n zeta (n,m) where qn is a rational number, the value m is held fixed, and (s,m) is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.For integer m, one has x=sum_{n=2}^infty q_n left[zeta(n)- sum_{k=1}^{m-1} k^{-n}right]. For m=2, a number of interesting numbers have a simple expression as rational zeta series: 1=sum_{n=2}^infty left[zeta(n)-1right] and 1-gamma=sum_{n=2}^infty frac{1}{n}left[zeta(n)-1right] where is the Euler-Mascheroni constant.

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