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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.

##### MSC:
 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
##### Citations:
Zbl 0032.01701; Zbl 1213.11146; Zbl 1153.11318
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