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

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
Full Text: