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Reciprocal Fibonacci constant
Mathematical constant defined as the sum of the reciprocals of the Fibonacci numbers

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

ψ = ∑ k = 1 ∞ 1 F k = 1 1 + 1 1 + 1 2 + 1 3 + 1 5 + 1 8 + 1 13 + 1 21 + ⋯ . {\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

ψ = 3.359885666243177553172011302918927179688905133732 … {\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(k 2) 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:

ψ = [ 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 , … ] {\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).

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In analogy to the Riemann zeta function, define the Fibonacci zeta function as ζ F ( s ) = ∑ n = 1 ∞ 1 ( F n ) s = 1 1 s + 1 1 s + 1 2 s + 1 3 s + 1 5 s + 1 8 s + ⋯ {\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 ζF (2s) is transcendental for any positive integer s, which is similar to the case of even-index Riemann zeta-constants ζ(2s).45
  • The constants ζF (2), ζF (4) and ζF (6) are algebraically independent.67
  • Except for ζF (1) which was proved to be irrational, the number-theoretic properties of ζF (2s + 1) (whenever s is a non-negative integer) are mostly unknown.8

See also

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. /wiki/Bill_Gosper ↩

  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 http://gallica.bnf.fr/ark:/12148/bpt6k5686125p/f9.image ↩

  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 978-93-80250-49-6 ↩

  4. 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 978-93-80250-49-6 ↩

  5. Waldschmidt, Michel (January 2022). "Transcendental Number Theory: recent results and open problems" (Lecture slides). /wiki/Michel_Waldschmidt ↩

  6. 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 978-93-80250-49-6 ↩

  7. Waldschmidt, Michel (January 2022). "Transcendental Number Theory: recent results and open problems" (Lecture slides). /wiki/Michel_Waldschmidt ↩

  8. 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 978-93-80250-49-6 ↩