Reciprocal Fibonacci constant

The reciprocal Fibonacci constant $ψ$ is the sum of the reciprocals of the Fibonacci numbers:

\psi =\sum _{k=1}^{\infty }{\frac {1}{F_{k}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+{\frac {1}{21}}+\cdots .

Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.

The value of $ψ$ is approximately

\psi =3.359885666243177553172011302918927179688905133732\dots

(sequence A079586 in the OEIS).

With $k$ terms, the series gives $O(k)$ digits of accuracy. Bill Gosper derived an accelerated series which provides $O(k 2)$ digits.^[1] $ψ$ is irrational, as was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.^[2]

Its simple continued fraction representation is:

\psi =[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2,\dots ]\!\,

(sequence A079587 in the OEIS).

Generalization and related constants

In analogy to the Riemann zeta function, define the Fibonacci zeta function as $\zeta _{F}(s)=\sum _{n=1}^{\infty }{\frac {1}{(F_{n})^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{5^{s}}}+{\frac {1}{8^{s}}}+\cdots$ for complex number $s$ with $\operatorname {Re} (s)>0$ , and its analytic continuation elsewhere. Particularly the given function equals $\psi$ when $s=1$ .^[3]

It was shown that:

The values of $\zeta _{F}(2s)$ are all transcendental for any positive integers $s$ , which is similar to the case of even-index Riemann zeta-constants $\zeta (2s)$ .^[3]^[4]
The constants $\zeta _{F}(2)$ , $\zeta _{F}(4)$ and $\zeta _{F}(6)$ are algebraically independent.^[3]^[4]
Except for $\zeta _{F}(1)$ which was proved to be irrational, the number-theoretic properties of $\zeta _{F}(2s+1)$ (whenever s is a non-negative integer) are mostly unknown.^[3]