On the irrationality of \(\zeta_q(2)\). (English. Russian original) Zbl 1109.11316

Zudilin, V. V.

Russ. Math. Surv. 56, No. 6, 1183-1185 (2001); translation from Usp. Mat. Nauk 56, No. 6, 147-148 (2001).

From the text: For complex \(q\), \(|q| < 1\), we define the quantity \[ \zeta_q(2):=\sum^\infty_{n=1}\;\frac{q^n}{(1-q^n)^2}=\sum^\infty_{n=1}\sigma(n)q^n;\quad \lim_{{q\to 1}\atop{|q|<1}}(1-q)^2\zeta_q(2) = \frac{\pi^2}{6}\,, \] where \(\sigma(n)\) is the sum of divisors of the positive integer \(n\). The following is proved: Theorem 1. When \(q = 1/p\), \(p\in \mathbb Z\setminus\{0,\pm1\}\), the number \(\zeta_q(2)\) is irrational and its index of irrationality satisfies the inequality \(\mu(\zeta_q(2))\leq 4.07869374\dots\).
The \(q\)-arithmetic scheme and the \(q\)-hypergeometric construction of approximating linear forms also enable us to sharpen the known measures of irrationality for the quantities \[ \zeta_q(1)=\sum^\infty_{n=1}\;\frac{q^n}{1-q^n}\,,\quad \ln_q(2)=\sum^\infty_{n=1}\;\frac{(-1)^{n-1}q^n}{1-q^n}\,,\quad |q|<1, \] which are the \(q\)-analogues of the (divergent) harmonic series and \(\log 2\), respectively.
Theorem 2. For \(q = 1/p\), \(p\in\mathbb Z\setminus\{0,\pm 1\}\), the indices of irrationality of the numbers (3) satisfy the inequalities \(\mu(\zeta_q(1))\leq 2.49846482\dots\), \(\mu(\ln_q(2))\leq 3.29727451\dots\).

MSC:

11J82	Measures of irrationality and of transcendence
11M41	Other Dirichlet series and zeta functions