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Precise estimates for the solution of Ramanujan’s generalized modular equation

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Abstract

In the article, we present several monotonicity theorems and inequalities for the modular equation functions \(m_{a}(r)\) and \(\mu _{a}(r),\) and find the infinite-series formulas for \(m_{1/3}(r)\) and \(m_{1/4}(r)\) which depend only on r. As applications, we find several precise explicit estimates for the solution of Ramanujan’s generalized modular equation.

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Acknowledgements

The authors express their sincere thanks to the referee(s) for careful reading of the manuscript and very helpful suggestions that improved the current manuscript substantially.

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Authors and Affiliations

  1. Department of Mathematics, Huzhou University, Huzhou, 313000, China

    Miao-Kun Wang

  2. College of Science, Hunan City University, Yiyang, 413000, China

    Yu-Ming Chu

  3. Friedman Brain Institute, Icahn School of Medicine at Mount Sinai, New York, NY, 10029, USA

    Wen Zhang

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  1. Miao-Kun Wang
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  2. Yu-Ming Chu
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Correspondence to Yu-Ming Chu.

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This research was supported by the Natural Science Foundation of China (Grant Nos. 61373169, 11701176, 11601485 ), the Science and Technology Research Program of Zhejiang Educational Committee (Grant No. Y201635325).

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Wang, MK., Chu, YM. & Zhang, W. Precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J 49, 653–668 (2019). https://doi.org/10.1007/s11139-018-0130-8

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