The irrationality of log(1+1/𝑞)log(1-1/𝑞)

Hata, Masayoshi

doi:10.1090/S0002-9947-98-01764-4

The irrationality of $\log (1+1/q) \log (1-1/q)$
HTML articles powered by AMS MathViewer

by Masayoshi Hata

Trans. Amer. Math. Soc. 350 (1998), 2311-2327

DOI: https://doi.org/10.1090/S0002-9947-98-01764-4

PDF | Request permission

Abstract:

We shall show that the numbers $1, \log (1+ 1/q), \log (1-1/q)$ and $\log (1+1/q)\log (1-1/q)$ are linearly independent over $\mathbf {Q}$ for any natural number $q \ge 54$. The key is to construct explicit Padé-type approximations using Legendre-type polynomials.

References

F. Beukers, A note on the irrationality of $\zeta (2)$ and $\zeta (3)$, Bull. London Math. Soc. 11 (1979), no. 3, 268–272. MR 554391, DOI 10.1112/blms/11.3.268
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
A. I. Galočkin, Lower bounds of polynomials in the values of a certain class of analytic functions, Mat. Sb. (N.S.) 95(137) (1974), 396–417, 471 (Russian). MR 0357338
—, a private communication.
Masayoshi Hata, Legendre type polynomials and irrationality measures, J. Reine Angew. Math. 407 (1990), 99–125. MR 1048530, DOI 10.1515/crll.1990.407.99
Masayoshi Hata, On the linear independence of the values of polylogarithmic functions, J. Math. Pures Appl. (9) 69 (1990), no. 2, 133–173. MR 1067449
Masayoshi Hata, Rational approximations to the dilogarithm, Trans. Amer. Math. Soc. 336 (1993), no. 1, 363–387. MR 1147401, DOI 10.1090/S0002-9947-1993-1147401-5
Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR 618278
G. Rhin and C. Viola, On the irrationality measure of $\zeta (2)$, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 1, 85–109 (English, with English and French summaries). MR 1209696