Irrationality measures for certain \(q\)-mathematical constants. (English) Zbl 1153.11034

For \(q\in\mathbb C\) with \(| q| >1\) define
\[ \pi_q=1+4\sum_{n=1}^\infty \frac {(-1)^{n-1}} {q^{2n-1}-1} \cdot p \]
S. Chowla [Proc. Nat. Inst. Sci. India, Part A 13, 171–173 (1947; Zbl 1153.11318)] and P. Erdős [J. Indian Math. Soc., II. Ser. 12, 63–66 (1948; Zbl 0032.01701)] showed that \(\pi_q\) is irrational when \(q\in\mathbb Z\setminus\{0,1,-1\}\). The authors [“Rational approximations to a \(q\)-analogue of \(\pi\) and some other \(q\)-series”, Diophantine approximation. Festschrift for Wolfgang Schmidt. Wien: Springer, Dev. Math. 16, 123–139 (2008; Zbl 1213.11146)] gave an irrationality measure for these \(\pi_q\): they proved with \(\mu=10.317\dots\) that the inequality \[ \left| \pi_q-\frac{P}{Q}\right| <Q^{-\mu} \] has only finitely many solutions in rational numbers \(P/Q\). Here they refine their estimate and prove the same result with the sharper value \(\mu=6.503\dots\) They deduce similar results for the numbers \[ \lambda_q= \sum_{n=1}^\infty \frac {1} {q^{2n-1}-1} \quad\text{and}\quad \beta_q= \sum_{n=1}^\infty \frac {(-1)^{n-1}} {q^n+1} \] with the exponents \(3.898\dots\) and \(3.940\dots\) respectively.


11J82 Measures of irrationality and of transcendence
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
30B50 Dirichlet series, exponential series and other series in one complex variable
Full Text: DOI