MATHEMATICS OF COMPUTATION Volume 75, Number 254, Pages 879–889 S 0025-5718(05)01812-0 Article electronically published on December 20, 2005 NEW IRRATIONALITY MEASURES FOR q-LOGARITHMS TAPANI MATALA-AHO, KEIJO V ¨ A ¨ AN ¨ ANEN, AND WADIM ZUDILIN Abstract. The three main methods used in diophantine analysis of q-series are combined to obtain new upper bounds for irrationality measures of the values of the q-logarithm function ln q (1 − z)= ∞  ν=1 z ν q ν 1 − q ν , |z|  1, when p =1/q ∈ Z \{0, ±1} and z ∈ Q. 1. Introduction The main purpose of this article is to improve the earlier irrationality measures of the values of the q-logarithm function (1) ln q (1 − z)= ∞  ν=1 z ν q ν 1 − q ν , |z|  1. In order to improve the earlier results we shall combine the following three major methods used in diophantine analysis of q-series: (1) a general hypergeometric construction of rational approximations to the values of q-logarithms vs. the q-arithmetic approach ([Z1]); (2) a continuous iteration procedure for additional optimization of analytic estimates ([Bo], [MV]); (3) introducing the cyclotomic polynomials for sharpening least common mul- tiples of the constructed linear forms in the case when z is a root of unity ([BV], [As], [MP]). Also, some standard analytic tools (i.e., from [Ha]) for deducing irrationality mea- sures will be required. We underline that in the corresponding arithmetic study of the values of the ordinary logarithm (cf. [Ru] for log 2 and [Ha] for other log- arithms) only feature (1) is mainly applied, but in particular feature (3) has no ordinary analogues. Thus the present q-problems invoke new attractions in arith- metic questions. We present the bounds for irrationality measures by means of certain estimates for irrationality exponents. Recall that the irrationality exponent of a real irrational Received by the editor June 16, 2004 and, in revised form, March 10, 2005. 2000 Mathematics Subject Classification. Primary 11J82, 33D15. This work is supported by an Alexander von Humboldt research fellowship and partially sup- ported by grant no. 03-01-00359 of the Russian Foundation for Basic Research. c 2005 American Mathematical Society 879