Mathematical constant
The reciprocal Fibonacci constant ψ 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 }
A079586 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:
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 ]\!\,}
A079587 OEIS ).
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 }
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:
See also
References
^ Gosper, William R. (1974), Acceleration of Series Massachusetts Institute of Technology , p. 66, hdl :1721.1/6088 ^ 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 ^ a b c d 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 ^ a b Waldschmidt, Michel (January 2022). "Transcendental Number Theory: recent results and open problems" (Lecture slides).
External links
![{\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 ]\!\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c66dc6fb55a42af1ae437279f1720b45fcbc49)
