Carsten Elsner

Elsner, Luca and Tachiya \cite{luca} proved that the values of the Jacobi-theta constants $\theta_3(m\tau)$ and $\theta_3(n\tau)$ are algebraically independent over ${\Q}$ for distinct integers $m,n$ under some conditions on $\tau$. On the other hand, in 2018 Elsner and Tachiya \cite{elsner} also proved that three values $\theta_3(m\tau),\theta_3(n\tau)$ and $\theta_3(\ell \tau)$ are algebraically dependent over ${\Q}$. In this article we prove the non-vanishing of linear forms in $\theta_3(m\tau)$, $\theta_3(n\tau)$ and $\theta_3(\ell \tau)$ under various conditions on $m,n,\ell$, and $\tau$. Among other things we prove that for odd and distinct positive integers $m,n>3$ the three numbers $\theta_3(\tau)$, $\theta_3(m\tau)$ and $\theta_3(n \tau)$ are linearly independent over $\overline{\mathbb{Q}}$ when $\tau$ is an algebraic number of some degree greater or equal to 3. In some sense this fills the gap between the above-mentioned former results on theta constants. A theorem on the linear independence over $\mathbb{C(\tau)}$ of the functions $\theta_3(a_1 \tau),\ldots,\theta_3(a_m \tau)$ for distinct positive integers $a_1, \ldots a_m$ is also established.

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.

Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by \[\zeta_F(s) \,:=\, \sum_{n=1}^{\infty} \frac{1}{F_n^s}\,,\quad \zeta_F^*(s) \,:=\,\sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{F_n^s}\,,\quad \zeta_L(s) \,:=\, \sum_{n=1}^{\infty} \frac{1}{L_n^s}\,,\quad \zeta_L^*(s) \,:=\, \sum_{n=1}^{\infty} \frac{{(-1)}^{n+1}}{L_n^s} \,.\] For positive integers $s$ the transcendence of these values is known as well as algebraic independence or dependence results for sets of these numbers. 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 \[\sum_{s=1}^m \big( \,t_s\zeta_F(2s) + u_s\zeta_F^*(2s) + v_s\zeta_L(2s) + w_s\zeta_L^*(2s) \,\big) \,=\, 0 \] 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 M. Stein, who, in 2012, was inspired by the relation $-2\zeta_F(2)+\zeta_F^*(2)+5\zeta_L^*(2)=0$. All the results are also true for zeta functions in $s$ where the Fibonacci and Lucas numbers are replaced by numbers from sequences satisfying a second order recurrence formula.

Let $\theta_3(\tau)=1+2\sum_{\nu=1}^{\infty} q^{\nu^2}$ with $q=e^{i\pi \tau}$ and $\Im (\tau)>0$ denote the Thetanullwert of the Jacobi theta function \[\theta(z|\tau) \,=\,\sum_{\nu=-\infty}^{\infty} e^{\pi i\nu^2\tau + 2\pi i\nu z} \,.\] Moreover, let $\theta_2(\tau)=2\sum_{\nu=0}^{\infty} q^{{(\nu+1/2)}^2}$ and $\theta_4(\tau)=1+2\sum_{\nu=1}^{\infty} {(-1)}^{\nu}q^{\nu^2}$. For every even integer $n\geq 6$, which is not a power of two, we prove constructively the existence of a nontrivial integer polynomial $Q_n(X,Y)$ such that \[Q_n\Big( \,\frac{\theta_3^4(n\tau)}{\theta_3^4(\tau)},\frac{\theta_2^4(\tau)}{\theta_3^4(\tau)}\, \Big) \,=\, 0 \] holds for all complex numbers $\tau$ from the upper half plane of $\mathbb{C}$. These polynomials are used to prove the algebraic independence of $\theta_3(n\tau)$ and $\theta_3(\tau)$ for all algebraic numbers $q=e^{i\pi \tau}$ with $0<|q|<1$. Combining this with former results of the authors, it is shown that for such algebraic $q$ the numbers $\theta_3(n\tau)$ and $\theta_3(\tau)$ are algebraically independent over $\mathbb{Q}$ for every integer $n\geq 2$. A result on the algebraic dependence over $\mathbb{Q}$ of the three numbers $\theta_3(\ell\tau)$, $\theta_3(m\tau)$, and $\theta_3(n\tau)$ for integers $\ell,m,n\geq 1$ is also presented.

Let $\theta_3(\tau)=1+2\sum_{\nu=1}^{\infty} q^{\nu^2}$ with $q=e^{i\pi \tau}$ denote the Thetanullwert of the Jacobi theta function \[\theta(z|\tau) \,=\,\sum_{\nu=-\infty}^{\infty} e^{\pi i\nu^2\tau + 2\pi i\nu z} \,.\] Moreover, let $\theta_2(\tau)=2\sum_{\nu=0}^{\infty} q^{{(\nu+1/2)}^2}$ and $\theta_4(\tau)=1+2\sum_{\nu=1}^{\infty} {(-1)}^{\nu}q^{\nu^2}$. For algebraic numbers $q$ with $0<|q|<1$ and for any $j\in \{ 2,3,4\}$ we prove the algebraic independence over $\mathbb{Q}$ of the numbers $\theta_j(n\tau)$ and $\theta_j(\tau)$ for all odd integers $n\geq 3$. Assuming the same conditions on $q$ and $\tau$ as above, we obtain sufficient conditions by use of a criterion involving resultants in order to decide on the algebraic independence over $\mathbb{Q}$ of $\theta_j(2m\tau)$ and $\theta_j(\tau)$ $(j=2,3,4)$ and of $\theta_3(4m\tau)$ and $\theta_3(\tau)$ with odd positive integers $m$. In particular, we prove the algebraic independence of $\theta_3(n\tau)$ and $\theta_3(\tau)$ for even integers $n$ with $2\leq n\leq 22$. The paper continues the work of the first-mentioned author, who already proved the algebraic independence of $\theta_3(2^m\tau)$ and $\theta_3(\tau)$ for $m=1,2,\dots$.