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A MODULAR PROOF OF TWO OF RAMANUJAN’S FORMULAE FOR $1/\unicode[STIX]{x1D70B}$

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 February 2019

YUE ZHAO Open the ORCID record for YUE ZHAO [Opens in a new window]
YUE ZHAO*
Affiliation:
ECSE Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA email [email protected]
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Abstract

In this article, we give new proofs of two of Ramanujan’s $1/\unicode[STIX]{x1D70B}$ formulae

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2\sqrt{2}}{99^{2}}\mathop{\sum }_{m=0}^{\infty }(26390m+1103)\frac{(4m)!}{396^{4m}(m!)^{4}}\end{eqnarray}$$
and
$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2}{84^{2}}\mathop{\sum }_{m=0}^{\infty }(21460m+1123)\frac{(-1)^{m}(4m)!}{(84\sqrt{2})^{4m}(m!)^{4}}\end{eqnarray}$$
 using the theory of modular forms. The method can also be used to prove other classical $1/\unicode[STIX]{x1D70B}$ formulae.

Keywords

Ramanujan seriestheta seriesDedekind eta functionsmodular forms

MSC classification

Primary: 11F03: Modular and automorphic functions
Secondary: 11F20: Dedekind eta function, Dedekind sums 11F27: Theta series; Weil representation; theta correspondences
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 1 , August 2020 , pp. 131 - 144
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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References

Borcherds, R. E., ‘Reflection groups of Lorentzian lattices’, Duke Math. J. 104 (2000), 319–366.Google Scholar
Borwein, J. M. and Borwein, P. B., Pi and the AGM (Wiley, New York, 1987).Google Scholar
Chan, H. H. and Cooper, S., ‘Rational analogues of Ramanujan’s series for 1/𝜋’, Math. Proc. Cambridge Philos. Soc. 153(2) (2012), 361–383.Google Scholar
Diamond, F. and Shurman, J., A First Course in Modular Forms, Graduate Texts in Mathematics, 228 (Springer, New York, 2005).Google Scholar
Ligozat, G., ‘Courbes modulaires de genre 1’, Mém. Soc. Math. Fr. 43 (1975), 7–80.Google Scholar
Mazur, B. and Swinnerton-Dyer, P., ‘Arithmetic of Weil curves’, Invent. Math. 25(1) (1974), 1–61.Google Scholar
Newman, M., ‘Construction and application of a class of modular functions’, Proc. Lond. Math. Soc. 3(1) (1957), 334–350.Google Scholar
Ramanujan, S., ‘Modular equations and approximations to Pi’, Q. J. Pure Appl. Math. 45 (1914), 350–372.Google Scholar
Weber, H. M., Lehrbuch der Algebra, Vol. III (Chelsea, New York, 1961).Google Scholar
Zhao, Y., ‘Chudnovsky’s formula for $1/\unicode[STIX]{x1D70B}$ revisited’, Preprint, 2018, arXiv:1807.10125.Google Scholar
Zudilin, W., ‘Ramanujan-type formulae for 1/𝜋: a second wind?’, in: Modular Forms and String Duality (eds. Yui, N., Verrill, H. and Doran, C. F.) (American Mathematical Society, Providence, RI, 2008), 179–188.Google Scholar