On an approximate identity of Ramanujan

Zagier, Don

doi:10.1007/BF02837833

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On an approximate identity of Ramanujan

Zagier, D. (1987). On an approximate identity of Ramanujan. Proceedings of the Indian Academy of Sciences. Mathematical Sciences, 97(1-3), 313-324. doi:10.1007/BF02837833.

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Item Permalink: https://hdl.handle.net/21.11116/0000-0004-395D-5 Version Permalink: https://hdl.handle.net/21.11116/0000-0004-395E-4

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Zagier, Don¹, Author

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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201

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Abstract: In his second notebook, Ramanujan wrote that 1-qx/(1+q\\sp 2/(1-q\\sp 3x/(1+q\\sp 4/(1-q\\sp 5x/(1+.. = q/(x+q\\sp 4/(x+q\\sp 8/(x+q\\sp12/(x+..\\quad nearly. This paper explains the relationship between these continued fractions and shows that for xgt;0, their difference is asymptotically \\exp (-c(x)/(1-q)) as q→ 1\\sp- where c(x) is monotone decreasing with c(0)=π\\sp 2/4, c(1)=π\\sp 2/5, and in general c(x) can be expressed in terms of the dilogarithm.

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Dates: Date issued: 1987

Publication Status: Issued

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Rev. Type: Internal

Identifiers: eDoc: 744949
Other: 111
DOI: 10.1007/BF02837833
URI: https://doi.org/10.1007/BF02837833

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Title: Proceedings of the Indian Academy of Sciences. Mathematical Sciences

Source Genre: Journal

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Publ. Info: Indian Academy of Sciences, Bangalore, Karnataka, India / Springer India, New Delhi, Delhi, India (distribution outside India)

Pages: - Volume / Issue: 97 (1-3) Sequence Number: - Start / End Page: 313 - 324 Identifier: ISSN: 0253-4142