Transcendence of Various Infinite Series and Applications of Baker's Theorem

Abstract

We consider various infinite series and examine their arithmetic nature. Series of interest are of the form $$\sum_{n =0}^{\infty} \frac{f(n)A(n)}{B(n)}, \;\;\;\; \sum_{n \in \mathbb{Z}}\frac{f(n)A(n)}{B(n)}, \;\;\;\; \sum_{n=0}^{\infty} \frac{z^n A(n)}{B(n)}$$ where $f$ is algebraic valued periodic function, $A(x), B(x) \in \overline{\mathbb{Q}}[x]$ and $z$ is an algebraic number with $|z| \leq 1$. We also examine multivariable extensions $$\sum_{n_1, \ldots, n_k = 0}^{\infty} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}$$ and $$\sum_{n_1, \ldots, n_k \in \mathbb{Z}} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}.$$

These series are all very natural things to write down and we would like to understand them better. We calculate closed forms using various techniques. For example, we use relations between Hurwitz zeta functions, digamma functions, polygamma functions, Fourier analysis, discrete Fourier transforms, among other objects and techniques. Once closed forms are found, we make use of some of the well-known transcendental number theory including the theorem of Baker regarding linear forms in logarithms of algebraic numbers to determine their arithmetic nature.

In one particular setting, we extend the work of Bundschuh \cite{bundschuh} by proving the following series are all transcendental for positive $c \in \mathbb{Q} \setminus \mathbb{Z}$ and $k$ a positive integer: $$\sum_{n \in \mathbb{Z}} \frac{1}{(n^2 + c)^k}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^4 - c^4)^{2k}}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^6 - c^6)^{2k}}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^3 \pm c^3)^{2k}}$$

$$\sum_{n \in \mathbb{Z}} \frac{1}{n^3 \pm c^3}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^3 -1}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^4 -1}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^6 -1}.$$ Bundschuh conjectured that the last three series are transcendental, but we offer the first unconditional proofs of transcendence.

We also show some conditional results under the assumption of some well-known conjectures. In particular, for $A_i(x), B_i(x) \in \overline{\mathbb{Q}}[x]$ with each $B_i(x)$ has only simple rational roots, if Schanuel's conjecture is true, the series (avoiding roots of the denominator) $$\mathop{{\sum}}_{n_1, \ldots, n_k =0}^{\infty} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}$$ is either an effectively computable algebraic number or transcendental.

We also show that Schanuel's conjecture implies that the series $$\sum_{n \in \mathbb{Z}} \frac{A(n)}{B(n)}$$ is either zero or transcendental, when $B(x)$ has non-integral roots.

We develop a general theory, analyzing various infinite series throughout.