show/hide this revision's text 3 corrected misspeling, added links; deleted 1 characters in body

To accomplish 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]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 difficuluty 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.
show/hide this revision's text 2 improved

To accomplish Joseph's answer, I add my review MR2354148 on [C. Elsner, S. Shimomura and I. Shiokawa, Acta Arith.{\bf 130} :1 (2007), no. 1, 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, \emph{Sb. Math.} \textbf{187}:9 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, \emph{Math. Math. Notes} \textbf{72}:5-6 72:5-6 (2002), 858--862. MR1964151] and [C. Krattenthaler, T. Rivoal, and W. Zudilin, \emph{J. J. Inst. Math. Jussieu} \textbf{5}:1 5:1 (2006), 53--79. MR2195945]. In particular, it is natural to expect a `Fibonacci' "Fibonacci" analogue of Rivoal's theorem [T. Rivoal, \emph{C. C. R. Acad. Sci. Paris Ser. I Math.} \textbf{331}:4 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 difficuluty 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.
show/hide this revision's text 1

To accomplish Joseph's answer, I add my review MR2354148 on [C. Elsner, S. Shimomura and I. Shiokawa, Acta Arith. {\bf 130} (2007), no. 1, 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, \emph{Sb. Math.} \textbf{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, \emph{Math. Notes} \textbf{72}:5-6 (2002), 858--862. MR1964151] and [C. Krattenthaler, T. Rivoal, and W. Zudilin, \emph{J. Inst. Math. Jussieu} \textbf{5}:1 (2006), 53--79. MR2195945]. In particular, it is natural to expect a `Fibonacci' analogue of Rivoal's theorem [T. Rivoal, \emph{C. R. Acad. Sci. Paris Ser. I Math.} \textbf{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 difficuluty of proving the transcendence for odd $\zeta_F(s)$ with $s$ odd is similar to the one for odd zeta values.