TITLE: Reciprocals of Fibonacci numbers
QUESTION [18 upvotes]: Is the sum of the reciprocals of Fibonacci numbers a transcendental?

REPLY [12 votes]: To supplement Joseph's answer, I add my review MR2354148 on [C. Elsner, S. Shimomura and I. Shiokawa, Acta Arith. 130:1 (2007), 37--60].
Let $\lbrace F_n\rbrace _{n\ge0}$ and $\lbrace L_n\rbrace _{n\ge0}$ be Fibonacci and Lucas numbers,
respectively,
$F_0=0$, $F_1=1$, $F_{n+2}=F_n+F_{n+1}$ for $n\ge0$, and
$L_0=2$, $L_1=1$, $L_{n+2}=L_n+L_{n+1}$ for $n\ge0$.
Using Nesterenko's theorem
[Yu.V. Nesterenko, Sb. Math. 187:9 (1996), 1319--1348. MR1422383]
and expressing the series
$$
\zeta_F(2s)=\sum_{n=1}^\infty\frac1{F_n^{2s}}
\quad\text{and}\quad
\zeta_L(2s)=\sum_{n=1}^\infty\frac1{L_n^{2s}},
\qquad s=1,2,\dots,
\qquad (1)
$$
via the Eisenstein series
$$
E_{2s}(q)=1-\frac{4s}{B_{2s}}\sum_{n=1}^\infty\sigma_{2s-1}(n)q^n,
\qquad \sigma_k(n)=\sum_{d\mid n}d^k,
$$
where $B_{2s}\in\mathbb Q$ are Bernoulli numbers,
the authors prove the algebraic independence of
the numbers in the collections $\zeta_F(2)$, $\zeta_F(4)$, $\zeta_F(6)$
and $\zeta_L(2)$, $\zeta_L(4)$, $\zeta_L(6)$ as well as express
algebraically even "zeta values" $\zeta_F(2s)$ (and $\zeta_L(2s)$)
for $s\ge4$ in terms of the three algebraically independent numbers
in the corresponding collection. Similar algebraic independence results
are shown for the alternating versions of (1). Known irrationality
results for $\zeta_F(k)$ and $\zeta_L(k)$ with odd $k$
(when the series have no known relations with the modular world) are indicated.
It is worth mentioning that these results go in a natural parallel
with the ones for the so-called $q$-zeta values defined in
[W. Zudilin, Math. Notes 72:5-6 (2002), 858--862. MR1964151]
and [C. Krattenthaler, T. Rivoal, and W. Zudilin,
J. Inst. Math. Jussieu 5:1 (2006), 53--79. MR2195945].
In particular, it is natural to expect a "Fibonacci" analogue of Rivoal's theorem
[T. Rivoal, C. R. Acad. Sci. Paris Ser. I Math. 331:4 (2000), 267--270. MR1787183]
on the infiniteness of
irrational numbers in the set $\zeta_F(1),\zeta_F(3),\zeta_F(5),\dots$
(or $\zeta_L(1),\zeta_L(3),\zeta_L(5),\dots$),
based on the techniques developed in the paper under review and
in the joint paper of Krattenthaler, Rivoal and the reviewer cited above.
Reviewed by Wadim Zudilin

To summarize, the difficulty of proving the transcendence for odd $\zeta_F(s)$ with $s$ odd is similar to the one for odd zeta values. The irrationality of $\zeta_F(1)$ is known but already its non-quadraticity remains an open problem.