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.^[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:

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

Generalization and related constants

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.^[3]

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).^[3][4]
• The constants ζ_F(2), ζ_F(4) and ζ_F(6) are algebraically independent.^[3][4]
• 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.^[3]

See also

• List of sums of reciprocals
• List of mathematical constants

References

1. Gosper, William R. (1974), Acceleration of Series, Artificial Intelligence Memo #304, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, p. 66, hdl:1721.1/6088.
2. André-Jeannin, Richard (1989), "Irrationalité de la somme des inverses de certaines suites récurrentes", Comptes Rendus de l'Académie des Sciences, Série I, 308 (19): 539–541, MR 0999451
3. Murty, M. Ram (2013), "The Fibonacci zeta function", in Prasad, D.; Rajan, C. S.; Sankaranarayanan, A.; Sengupta, J. (eds.), Automorphic representations and L-functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 22, Tata Institute of Fundamental Research, pp. 409–425, ISBN 978-93-80250-49-6, MR 3156859
4. Waldschmidt, Michel (January 2022). "Transcendental Number Theory: recent results and open problems" (Lecture slides).

External links

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