Abstract
In 1968, Ramachandra [1, 2] proved results relating to the set of complex numbers at which a given set of algebraically independent meromorphic functions assumes values in a fixed algebraic number field. These results proved to be significant in the case, to quote his own words “(overlooked by Gelfond) where the functions concerned do not satisfy algebraic differential equations of the first order with algebraic number coefficients.” His result, besides simplifying Schneider’s method, enables one to study the set of all complex numbers at which two algebraically independent meromorphic functions f(z) and g(z) take values which are algebraic numbers. In particular, he was able to obtain results when \((f(z), g(z))\in \{(z,\wp (az)),(e^z,\wp (az)),(\wp _1(z),\wp _2(az))\}\) where \(a\ne 0\) is an arbitrary complex number and \(\wp ,\wp _1\) and \(\wp _2\) are Weierstrass elliptic functions. We refer to [2] for these results.
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References
K. Ramachandra, Contributions to the theory of transcendental numbers I. Acta Arithmetica 14, 65–72 (1968); II 14, 73–88 (1968)
K. Ramachandra, Lectures on Transcendental Numbers (The Ramanujan Institute, University of Madras, Chennai, 1969)
T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 41 (Springer, New York, 1990)
M. Ram Murty, P. Rath, Transcendental Numbers (Springer, Berlin, 2014), 217 pp
L. Alaoglu, P. Erdős, On highly composite and similar numbers. Trans. Amer. Math. Soc. 56, 448–469 (1944)
K. Senthil Kumar, R. Thangadurai, V. Kumar, On a problem of Alaoglu and Erdős. Resonance 23(7), 749–758 (2018)
M. Waldschmidt, Nombres Transcendants (Springer, Berlin, 1974)
T.N. Shorey, On the sum \(\displaystyle \sum _{k=1}^3\left|2^{\pi ^k}-\alpha _k\right|, \alpha _k\) algebraic numbers. J. Number Theory 6, 248–260 (1974)
S. Srinivasan, On algebraic approximation to \(2^{\pi ^k}\)\((k=1,2,3,\ldots )\), I. Indian J. Pure Appl. Math. 5, 513–523 (1974)
S. Srinivasan, On algebraic approximation to \(2^{\pi ^k}\)\((k=1,2,3,\ldots )\), II. J. Indian Math. Soc. (N.S.) 43 (1979); (1–4), 53–60 (1980)
Yu.V. Nesterenko, Algebraic Independence, vol. 14 (Tata Institute of Fundamental Research Publications, Mumbai, 2008), 157 pp
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Natarajan, S., Thangadurai, R. (2020). Extensions Due to Ramachandra. In: Pillars of Transcendental Number Theory. Springer, Singapore. https://doi.org/10.1007/978-981-15-4155-1_4
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DOI: https://doi.org/10.1007/978-981-15-4155-1_4
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