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Mat. Zametki, 1975, Volume 18, Issue 4, Pages 541–552 (Mi mz9968)  

This article is cited in 7 scientific papers (total in 7 papers)

Lower bounds of linear forms of values of $G$ functions

A. I. Galochkin

M. V. Lomonosov Moscow State University

Abstract: Lower bounds are obtained for linear forms of values of Siegel's $G$ functions. In particular, it is found that if $\alpha_1,…,\alpha_m$ are pairwise distinct nonzero rational numbers, then for any positive $\varepsilon$ and a natural $q>q_0(\varepsilon,\alpha_1,…,\alpha_m)$ we have for any nonzero set $(x_0,x_1,…,x_m)$ of integers the inequality
$$ |x_0+x_1\ln(1+\alpha_1q^{-1})+…+x_m\ln(1+\alpha_mq^{-1})|>q^{-\lambda}(h_1…h_m)^{-1-\varepsilon}, $$
where $h_i=\max(1,|x_i|)$, and $\lambda=\lambda(\varepsilon,\alpha_1,…,\alpha_m)$.

Full text: PDF file (1129 kB)

English version:
Mathematical Notes, 1975, 18:4, 911–917

Bibliographic databases:

UDC: 511
Received: 14.10.1974

Citation: A. I. Galochkin, “Lower bounds of linear forms of values of $G$ functions”, Mat. Zametki, 18:4 (1975), 541–552; Math. Notes, 18:4 (1975), 911–917

Citation in format AMSBIB
\Bibitem{Gal75}
\by A.~I.~Galochkin
\paper Lower bounds of linear forms of values of $G$ functions
\jour Mat. Zametki
\yr 1975
\vol 18
\issue 4
\pages 541--552
\mathnet{http://mi.mathnet.ru/mz9968}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=414499}
\zmath{https://zbmath.org/?q=an:0319.10039}
\transl
\jour Math. Notes
\yr 1975
\vol 18
\issue 4
\pages 911--917
\crossref{https://doi.org/10.1007/BF01153043}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. E. M. Nikishin, “On logarithms of natural numbers”, Math. USSR-Izv., 15:3 (1980), 523–530  mathnet  crossref  mathscinet  zmath  isi
    2. V. G. Sprindzhuk, “Achievements and problems in Diophantine approximation theory”, Russian Math. Surveys, 35:4 (1980), 1–80  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. E. M. Matveev, “Linear forms in the values of $G$-functions, and Diophantine equations”, Math. USSR-Sb., 45:3 (1983), 379–396  mathnet  crossref  mathscinet  zmath
    4. W. V. Zudilin, “On a measure of irrationality for values of $G$-functions”, Izv. Math., 60:1 (1996), 91–118  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. W. V. Zudilin, “Cancellation of factorials”, Sb. Math., 192:8 (2001), 1181–1207  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. T. G. Hessami Pilehrood, Kh. Hessami Pilehrood, “Lower bounds for linear forms in values of polylogarithms”, Math. Notes, 77:4 (2005), 573–579  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Zudilin W., “Approximations to -, Di- and Tri-Logarithms”, J. Comput. Appl. Math., 202:2 (2007), 450–459  crossref  isi
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