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Mat. Zametki, 1976, Volume 20, Issue 1, Pages 35–45 (Mi mz7823)  

This article is cited in 8 scientific papers (total in 8 papers)

The sharpening of the bounds on certain linear forms

A. I. Galochkin

M. V. Lomonosov Moscow State University

Abstract: Let $g_1,…,g_{m-1}$, $b$, $h_1,…,h_m$ be the integers from some imaginary quadratic field, $b\ne0$, $\max|g_i|=g$, $\max|h_j|=H\ne0$, $P_m(x)=x^m+g_{m-1}x^{m-1}+…+g_1x$, $P_m(x)\ne0$ for x =$x=1,2,…$,
$$ \psi(z)=1+\sum_{\nu=1}^\infty[\prod_{x=1}^\nu P_m(x)]^{-1}z^\nu. $$
Then
$$ |h_1\psi(\frac1b)+h_2\psi'(\frac1b)+…+h_m\psi^{(m-1)}(\frac1b)|>CH^{1-m}\{\frac{\ln\ln(H+2)}{\ln(H+2)}\}^\gamma, $$
where $\gamma=(m-1)^2g-(m-1)\operatorname{Re}g_{m-1}+m(m^2+m-4)/2$, and $C=C(b,m,g)>0$.

Full text: PDF file (621 kB)

English version:
Mathematical Notes, 1976, 20:1, 575–581

Bibliographic databases:

UDC: 511
Received: 23.10.1975

Citation: A. I. Galochkin, “The sharpening of the bounds on certain linear forms”, Mat. Zametki, 20:1 (1976), 35–45; Math. Notes, 20:1 (1976), 575–581

Citation in format AMSBIB
\Bibitem{Gal76}
\by A.~I.~Galochkin
\paper The sharpening of the bounds on certain linear forms
\jour Mat. Zametki
\yr 1976
\vol 20
\issue 1
\pages 35--45
\mathnet{http://mi.mathnet.ru/mz7823}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=424709}
\zmath{https://zbmath.org/?q=an:0336.10028}
\transl
\jour Math. Notes
\yr 1976
\vol 20
\issue 1
\pages 575--581
\crossref{https://doi.org/10.1007/BF01152761}


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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. V. G. Sprindzhuk, “Achievements and problems in Diophantine approximation theory”, Russian Math. Surveys, 35:4 (1980), 1–80  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. A. I. Galochkin, “On estimates, unimprovable with respect to height, of some linear forms”, Math. USSR-Sb., 52:2 (1985), 407–421  mathnet  crossref  mathscinet  zmath
    3. P. L. Ivankov, “On arithmetic properties of the values of hypergeometric functions”, Math. USSR-Sb., 72:1 (1992), 267–286  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. Hancl J., Leinonen M., Leppala K., Matala-aho T., “Explicit Irrationality Measures for Continued Fractions”, J. Number Theory, 132:8 (2012), 1758–1769  crossref  isi
    5. Dodulikova S. Hancl J. Kolouch O. Leinonen M. Leppala K., “Irrationality Measures For Almost Periodic Continued Fractions”, Georgian Math. J., 23:1 (2016), 55–67  crossref  isi
    6. Hancl J., Leppala K., “Irrationality Measures For Continued Fractions With Asymptotic Conditions”, Kyushu J. Math., 70:2 (2016), 205–216  crossref  mathscinet  zmath  isi  elib
    7. P. L. Ivankov, “O differentsirovanii po parametru gipergeometricheskoi funktsii spetsialnogo vida”, Izv. vuzov. Matem., 2019, no. 12, 71–81  mathnet  crossref
    8. P. L. Ivankov, “O znacheniyakh gipergeometricheskoi funktsii s parametrom iz kvadratichnogo polya”, Chebyshevskii sb., 20:2 (2019), 178–185  mathnet  crossref
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