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Modularity of a certain continued fraction of Ramanujan

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Abstract

We apply the methods of Lee and Park to study a certain continued fraction \(H(\tau )\) of Ramanujan, which is a particular case of his identity written in one of his notebooks. We prove that \(H(\tau )\) can be expressed in terms of an \(\eta \)-quotient \(s(\tau )\), which is a generator for the field of all modular functions on the congruence subgroup \(\Gamma _0(12)\). We also show that there is a modular equation for \(s(\tau )\) and \(H(\tau )\) of level n for all positive integers n, and provide explicitly the modular equations of level p for primes \(p\le 13\).

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References

  1. Aricheta, V.M.: Supersingular elliptic curves and moonshine. SIGMA Symm. Integr. Geom. Methods Appl. 15, 007 (2019)

    MathSciNet  Google Scholar 

  2. Berndt, B.C.: Ramanujan’s Notebooks. Part III. Springer, New York (1991)

    Book  Google Scholar 

  3. Biagioli, A.J.F.: The construction of modular forms as products of transforms of the Dedekind eta function. Acta Arith. 54, 273–300 (1990)

    Article  MathSciNet  Google Scholar 

  4. Cho, B., Koo, J.K.: Construction of class fields over imaginary quadratic fields and applications. Q. J. Math. 61, 199–216 (2010)

    Article  MathSciNet  Google Scholar 

  5. Cho, B., Koo, J.K., Park, Y.K.: Arithmetic of the Ramanujan–Göllnitz–Gordon continued fraction. J. Number Theory 129, 922–947 (2009)

    Article  MathSciNet  Google Scholar 

  6. Cho, B., Kim, N.M., Koo, J.K.: Affine models of the modular curves \(X(p)\) and its application. Ramanujan J. 24, 235–257 (2011)

    Article  MathSciNet  Google Scholar 

  7. Gee, A.: Class invariants by Shimura’s reciprocity law. J. Théor. Nombres Bordeaux 11, 45–72 (1999)

    Article  MathSciNet  Google Scholar 

  8. Gordon, B., Hughes, K.: Multiplicative properties of \(\eta\)-products II. Contemp. Math. 143, 415–430 (1993)

    Article  MathSciNet  Google Scholar 

  9. Ishida, N., Ishii, N.: The equations for modular function fields of principal congruence subgroups of prime level. Manuscr. Math. 90, 271–285 (1996)

    Article  MathSciNet  Google Scholar 

  10. Lee, Y., Park, Y.K.: Three Ramanujan continued fractions with modularity. J. Number Theory 188, 299–323 (2018)

    Article  MathSciNet  Google Scholar 

  11. Ligozat, G.: Courbes modulaires de genus 1. Mém. S. M. F. 43, 5–80 (1975)

    MathSciNet  Google Scholar 

  12. Martin, Y.: Multiplicative \(\eta \)-quotients. Trans. Am. Math. Soc. 348, 4825–4856 (1996)

    Article  MathSciNet  Google Scholar 

  13. Miranda, R.: Algebraic Curves and Riemann Surfaces. Graduate Studies in Mathematics, vol. 5. American Mathematical Society, Rhode Island (1995)

  14. Newman, M.: Construction and application of a class of modular functions. Proc. Lond. Math. Soc. s3–7, 334–350 (1957)

    Article  MathSciNet  Google Scholar 

  15. Newman, M.: Construction and application of a class of modular functions (II). Proc. Lond. Math. Soc. 3–9, 373–387 (1959)

    Article  MathSciNet  Google Scholar 

  16. Park, Y.K.: Analogue of Ramanujan’s function \(k(\tau)\) for the cubic continued fraction. Int. J. Number Theory. 19, 2101–2120 (2023)

    Article  MathSciNet  Google Scholar 

  17. Saikia, N.: Modular identities and explicit evaluations of a continued fraction of Ramanujan. Int. J. Math. Math. Sci. 2012(694251), 1–10 (2012)

    Article  MathSciNet  Google Scholar 

  18. Saikia, N.: A new continued fraction of Ramanujan, its modular identities and explicit evaluations. Afr. Mat. 26, 407–417 (2013)

    Article  MathSciNet  Google Scholar 

  19. Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton (1971)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank his PhD supervisor Dr. Victor Manuel Aricheta for his guidance in this work. The author would also like to thank the anonymous referee for helpful comments that improved the contents of the manuscript.

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  1. Institute of Mathematics, University of the Philippines, Diliman, 1101, Quezon City, Philippines

    Russelle Guadalupe

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  1. Russelle Guadalupe
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Correspondence to Russelle Guadalupe.

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Guadalupe, R. Modularity of a certain continued fraction of Ramanujan. Ramanujan J 63, 947–967 (2024). https://doi.org/10.1007/s11139-023-00795-1

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  • DOI: https://doi.org/10.1007/s11139-023-00795-1

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