Reciprocal Fibonacci constant

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

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

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

$${\displaystyle \psi =3.359885666243177553172011302918927179688905133732\dots }$$(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(k²) digits. ψ is irrational, as was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.

Its simple continued fraction representation is:

$${\displaystyle \psi =[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2,\dots ]\!\,}$$(sequence A079587 in the OEIS).

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

It was shown that:

• The value of ζ_F(2s) is transcendental for any positive integer s, which is similar to the case of even-index Riemann zeta-constants ζ(2s).
• The constants ζ_F(2), ζ_F(4) and ζ_F(6) are algebraically independent.
• Except for ζ_F(1) which was proved to be irrational, the number-theoretic properties of ζ_F(2s + 1) (whenever s is a non-negative integer) are mostly unknown.

• List of sums of reciprocals
• List of mathematical constants

• Weisstein, Eric W. "Reciprocal Fibonacci Constant". MathWorld.