Let $F_{n}$ and $L_{n}$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by $$\begin{eqnarray}\unicode[STIX]{x1D701}_{F}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{F}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{L_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{L_{n}^{s}}}.\end{eqnarray}$$ As a consequence of Nesterenko’s proof of the algebraic independence of the three Ramanujan functions $R(\unicode[STIX]{x1D70C}),Q(\unicode[STIX]{x1D70C}),$ and $P(\unicode[STIX]{x1D70C})$ for any algebraic number $\unicode[STIX]{x1D70C}$ with $0<\unicode[STIX]{x1D70C}<1$, the algebraic independence or dependence of various sets of these numbers is already known for positive even integers $s$. In this paper, we investigate linear forms in the above zeta functions and determine the dimension of linear spaces spanned by such linear forms. In particular, it is established that for any positive integer $m$ the solutions of $$\begin{eqnarray}\mathop{\sum }_{s=1}^{m}(t_{s}\unicode[STIX]{x1D701}_{F}(2s)+u_{s}\unicode[STIX]{x1D701}_{F}^{\ast }(2s)+v_{s}\unicode[STIX]{x1D701}_{L}(2s)+w_{s}\unicode[STIX]{x1D701}_{L}^{\ast }(2s))=0\end{eqnarray}$$ with $t_{s},u_{s},v_{s},w_{s}\in \mathbb{Q}$$(1\leq s\leq m)$ form a $\mathbb{Q}$-vector space of dimension $m$. This proves a conjecture from the Ph.D. thesis of Stein, who, in 2012, was inspired by the relation $-2\unicode[STIX]{x1D701}_{F}(2)+\unicode[STIX]{x1D701}_{F}^{\ast }(2)+5\unicode[STIX]{x1D701}_{L}^{\ast }(2)=0$. All the results are also true for zeta functions in $2s$, where the Fibonacci and Lucas numbers are replaced by numbers from sequences satisfying a second-order recurrence formula.

中文翻译：

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由二阶递归序列构成的狄利克莱级数的线性关系

让$F_{n}$和$L_{n}$分别是斐波那契数和卢卡斯数。四个对应的 zeta 函数$$$定义为$$\begin{eqnarray}\unicode[STIX]{x1D701}_{F}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{ F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{F}^{\ast }(s):=\mathop{\sum }_{n=1}^{\ infty }{\displaystyle \frac{(-1)^{n+1}}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}(s):= \mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{L_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L }^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{L_{n} ^{s}}}.\end{eqnarray}$$由于 Nesterenko 证明了三个 Ramanujan 函数的代数独立性$R(\unicode[STIX]{x1D70C}),Q(\unicode[STIX]{x1D70C}),$和$P(\unicode[STIX]{x1D70C})$对于任何代数数$\unicode[STIX]{x1D70C}$和$0<\unicode[STIX]{x1D70C}<1$，这些数的各种集合的代数独立性或相关性对于正偶数是已知的$$$. 在本文中，我们研究了上述 zeta 函数中的线性形式，并确定了由这些线性形式跨越的线性空间的维数。特别是，确定对于任何正整数$m$的解决方案$$\begin{eqnarray}\mathop{\sum }_{s=1}^{m}(t_{s}\unicode[STIX]{x1D701}_{F}(2s)+u_{s}\unicode [STIX]{x1D701}_{F}^{\ast }(2s)+v_{s}\unicode[STIX]{x1D701}_{L}(2s)+w_{s}\unicode[STIX]{x1D701 }_{L}^{\ast }(2s))=0\end{eqnarray}$$和$t_{s},u_{s},v_{s},w_{s}\in \mathbb{Q}$$(1\leq s\leq m)$形成一个$\mathbb{Q}$-维向量空间$m$. 这证明了博士的一个猜想。Stein 的论文，他在 2012 年受到关系的启发$-2\unicode[STIX]{x1D701}_{F}(2)+\unicode[STIX]{x1D701}_{F}^{\ast }(2)+5\unicode[STIX]{x1D701}_ {L}^{\ast }(2)=0$. 对于 zeta 函数，所有结果也是如此$2s$，其中斐波那契数和卢卡斯数被替换为满足二阶递归公式的序列中的数。