Reciprocal Fibonacci constant

The reciprocal Fibonacci constant is the sum of the reciprocals of the Fibonacci numbers:
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 is approximately
.
With terms, the series gives digits of accuracy. Bill Gosper derived an accelerated series which provides digits.
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:
.

Generalization and related constants

In analogy to the Riemann zeta function, define the Fibonacci zeta function as
for complex number with, and its analytic continuation elsewhere. Particularly the given function equals when.
It was shown that:

The value of is transcendental for any positive integer, which is similar to the case of even-index Riemann zeta-constants.
The constants, and are algebraically independent.
Except for which was proved to be irrational, the number-theoretic properties of are mostly unknown.