## Approximations to $$q$$-logarithms and $$q$$-dilogarithms, with applications to $$q$$-zeta values.(English)Zbl 1088.11052

J. Math. Sci., New York 137, No. 2, 4617-4633 (2006); translation from Zap. Nauchn. Semin. POMI 322, 107-124 (2005).
The author is able to construct explicit simultaneous approximations to the $$q$$-series $$L_1(x_i;q)$$ for $$x_i$$, $$i=1, 2, \dots,s$$, and also to the $$q$$-series $$L_1(x;q)$$ and $$L_2 (x;q)$$, where $L_1(x;q)=\sum_{n=1}^\infty {(xq)^n \over 1-q^n}=\sum_{n=1}^\infty {xq^n \over 1-xq^n}$ and $L_2(x;q)=\sum_{n=1}^\infty {n(xq)^n \over 1-q^n}=\sum_{n=1}^\infty {xq^n \over (1-xq^n)^2}.$ These constructions have arithmetical corrolaries, in particular a quantitative linear indepence over $${\mathbb Q}$$ for the three numbers $$1$$, $$\zeta_q(1)=L_1(1;q)$$ and $$\zeta_{q^2}(1)$$, and also for the three numbers $$1$$, $$\zeta_q(1)=L_1(1;q)$$ and $$\zeta_{q}(2)=L_2(1;q)$$ for $$q=1/p$$ when $$p\in {\mathbb Z}\setminus \{0,\pm1\}$$.

### MSC:

 11J82 Measures of irrationality and of transcendence 11M41 Other Dirichlet series and zeta functions 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
Full Text:

### References:

 [1] W. Van Assche, ”Little q-Legendre polynomials and irrationality of certain Lambert series,” Ramanujan J., 5, No. 3, 295–310 (2001). · Zbl 1035.11032 [2] J.-P. Bézivin, ”Indépendence linéaire des valeurs des solutions transcendantes de certaines équations fonctionelles,” Manuscripta Math., 61, 103–129 (1988). · Zbl 0644.10025 [3] P. Borwein, ”On the irrationality of $$\sum {({1 \mathord{\left/ {\vphantom {1 {(q^n + r)}}} \right. \kern-\nulldelimiterspace} {(q^n + r)}})}$$ ,” J. Number Theory, 37, 253–259 (1991). · Zbl 0718.11029 [4] P. Bundschuh and K. Vaananen, ”Arithmetical investigations of a certain infinite product,” Compositio Math., 91, 175–199 (1994). · Zbl 0802.11027 [5] P. Bundschuh and K. Vaananen, ”Linear independence of q-analogues of certain classical constants,” Results Math. (2005), to appear. [6] P. Bundschuh and W. Zudilin, ”Rational approximations to a q-analogue of {$$\pi$$} and some other q-series,” in: A 70th Birthday Conference in Honor of Wolfgang M. Schmidt (November 2003, Vienna), Berlin, Springer-Verlag (2005), to appear. · Zbl 1213.11146 [7] P. Erdos, ”On arithmetical properties of Lambert series,” J. Indiana Math. Soc., 12, 63–66 (1948). · Zbl 0032.01701 [8] G. Gasper and M. Rahman, ”Basic hypergeometric series,” Encyclopedia Mathematics, 35, Cambridge Univ. Press, Cambridge (1990). · Zbl 0695.33001 [9] M. Hata, ”Rational approximations to {$$\pi$$} and some other numbers,” Acta Arith., 63, No. 4, 335–349 (1993). · Zbl 0776.11033 [10] T. Matalo-aho, K. Väänänen, and W. Zudilin, ”New irrationality measures for q-logarithms,” Math. Comput.(2004), submitted. [11] K. Postelmans and W. Van Assche, Irrationality of {$$\zeta$$} q (1) and {$$\zeta$$} q (2), Manuscript (2004). [12] W. Zudilin, ”On the irrationality measure for a q-analog of {$$\zeta$$}(2),” Mat. Sb., 93, No. 8, 49–70 (2002). · Zbl 1044.11067 [13] W. Zudilin, ”Diophantine problems for q-zeta values,” Mat. Zametki, 72, No. 6, 936–940 (2002). · Zbl 1044.11066 [14] W. Zudilin, ”Heine’s basic transform and a permutation group for q-harmonic series,” Acta Arith., 111, No. 2, 153–164 (2004). · Zbl 1052.11053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.