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 = ∑ n = 2 ∞ q n ζ ( n , m ) {\displaystyle x=\sum _{n=2}^{\infty }q_{n}\zeta (n,m)}
where each qn is a rational number, the value m is held fixed, and ζ(s, m) is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.
Elementary series For integer m>1, one has
x = ∑ n = 2 ∞ q n [ ζ ( n ) − ∑ k = 1 m − 1 k − n ] {\displaystyle 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 = ∑ n = 2 ∞ [ ζ ( n ) − 1 ] {\displaystyle 1=\sum _{n=2}^{\infty }\left[\zeta (n)-1\right]}
and
1 − γ = ∑ n = 2 ∞ 1 n [ ζ ( n ) − 1 ] {\displaystyle 1-\gamma =\sum _{n=2}^{\infty }{\frac {1}{n}}\left[\zeta (n)-1\right]}
where γ is the Euler–Mascheroni constant. The series
log 2 = ∑ n = 1 ∞ 1 n [ ζ ( 2 n ) − 1 ] {\displaystyle \log 2=\sum _{n=1}^{\infty }{\frac {1}{n}}\left[\zeta (2n)-1\right]}
follows by summing the Gauss–Kuzmin distribution. There are also series for π:
log π = ∑ n = 2 ∞ 2 ( 3 / 2 ) n − 3 n [ ζ ( n ) − 1 ] {\displaystyle \log \pi =\sum _{n=2}^{\infty }{\frac {2(3/2)^{n}-3}{n}}\left[\zeta (n)-1\right]}
and
13 30 − π 8 = ∑ n = 1 ∞ 1 4 2 n [ ζ ( 2 n ) − 1 ] {\displaystyle {\frac {13}{30}}-{\frac {\pi }{8}}=\sum _{n=1}^{\infty }{\frac {1}{4^{2n}}}\left[\zeta (2n)-1\right]}
being notable because of its fast convergence. This last series follows from the general identity
∑ n = 1 ∞ ( − 1 ) n t 2 n [ ζ ( 2 n ) − 1 ] = t 2 1 + t 2 + 1 − π t 2 − π t e 2 π t − 1 {\displaystyle \sum _{n=1}^{\infty }(-1)^{n}t^{2n}\left[\zeta (2n)-1\right]={\frac {t^{2}}{1+t^{2}}}+{\frac {1-\pi t}{2}}-{\frac {\pi t}{e^{2\pi t}-1}}}
which in turn follows from the generating function for the Bernoulli numbers
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