Abstract
We study, from different points of view, the two series \(\chi_{+}(z)= \sum_{n\geq0} z^{2^{n}}\) and \(\chi_{-}(z)= \sum_{n\geq0}(-1)^{n} z^{2^{n}}\). We show that the first series is related to the Jacobi theta function and the second is related to the Dedekind eta function and to the modular curve X 0(14). We also present another approach to a celebrated identity of Hardy.
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Acknowledgements
I want to thank Roger Gay for the discussions we had on Ehrenpreis’ mathematics, Michel Mendès France for the discussions we had on Hardy’s identity and Nils-Peter Skoruppa for the discussions we had on Hecke’s theory.
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Sebbar, A. (2012). On Two Lacunary Series and Modular Curves. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_18
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DOI: https://doi.org/10.1007/978-88-470-1947-8_18
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