C-saddle Method and Beukers’ Integral

We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukers’ double integral. The two-dimensional version of the saddle method, which we call C2-saddle method, is applied. 0. Introduction F. Beukers [1] has introduced the following double and triple integrals: ∫∫ S L(x)(1 − y) 1− xy dx dy and ∫∫∫ B L(x)L(y) 1− u(1− xy) du dx dy, (0.1) giving elegant short proofs of the irrationality of ζ(2) = π/6 and ζ(3) respectively, where L(x) = (x(1 − x))/n! is the Legendre polynomial on the unit interval, S = [0, 1] and B = [0, 1]. These integrals are very important in the arithmetical study of ζ(2) and ζ(3), since there exist certain modifications of the integrands in (0.1), which produce fairly good irrationality measures for them. (See G. Rhin and C. Viola [10] and the author [7] for ζ(2), the author [4] for ζ(3).) The aim of this paper is to show that another modification of the double integral in (0.1) can produce good non-quadraticity measures for the values of logarithm at specific rational points. For example, it will be shown that there exists an effective constant H0 satisfying ∣∣log 2− ξ∣∣ ≥ H−25.0463 for any quadratic number ξ with H ≡ H(ξ) ≥ H0, where H(ξ) is the usual height of ξ, the maximum of absolute values of the coefficients of its minimal polynomial. (Shortly we say that log 2 has a non-quadraticity measure 25.0463.) Concerning non-quadraticity measures of log 2, H. Cohen [2] obtained the measure 287.819 by using some linear recurrence. Later E. Reyssat [9] obtained the measure 105 by considering the classical Padé approximation formula to logarithms. Our result mentioned above hence improves the earlier measures. Let us explain briefly how we modify Beukers’ integral. For any positive integer k, let Rk be the rectangular region (1, (k+ 1)/k)× (k/(k+ 1), 1). We then consider Received by the editors July 14, 1997 and, in revised form, August 26, 1998. 2000 Mathematics Subject Classification. Primary 11J82; Secondary 30E99.