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Odd values of the Ramanujan tau function

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Abstract

We prove a number of results regarding odd values of the Ramanujan \(\tau \)-function. For example, we prove the existence of an effectively computable positive constant \(\kappa \) such that if \(\tau (n)\) is odd and \(n \ge 25\) then either

$$\begin{aligned} P(\tau (n)) \; > \; \kappa \cdot \frac{\log \log \log {n}}{\log \log \log \log {n}} \end{aligned}$$

or there exists a prime \(p \mid n\) with \(\tau (p)=0\). Here P(m) denotes the largest prime factor of m. We also solve the equation \(\tau (n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}\) and the equations \(\tau (n)=\pm q^b\) where \(3\le q < 100\) is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey–Hellegouarch elliptic curves.

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References

  1. Balakrishnan, J.S., Craig, W., Ono, K.: Variations of Lehmer’s conjecture for Ramanujan’s tau-function, J. Number Theory (to appear)

  2. Balakrishnan, J.S., Craig, W., Ono, K., Tsai, W.-L.: Variants of Lehmer’s speculation for newforms. (submitted for publication)

  3. Barros, C.F.: On the Lebesgue-Nagell equation and related subjects, PhD thesis, University of Warwick, (2010)

  4. Bennett, M.A.: Rational approximation to algebraic numbers of small height : the Diophantine equation \(|ax^n-by^n|=1\). J. Reine Angew. Math. 535, 1–49 (2001)

    Article  MathSciNet  Google Scholar 

  5. Bennett, M.A., Skinner, C.: Ternary Diophantine equations via Galois representations and modular forms. Can. J. Math. 56(1), 23–54 (2004)

    Article  MathSciNet  Google Scholar 

  6. Bennett, M.A., Dahmen, S., Mignotte, M., Siksek, S.: Shifted powers in binary recurrence sequences. Math. Proc. Camb. Philos. Soc. 158, 305–329 (2015)

    Article  MathSciNet  Google Scholar 

  7. Bilu, Y., Hanrot, G., Voutier, P.: Existence of primitive divisors of Lucas and Lehmer numbers. J. Reine Angew. Math. 539, 75–122 (2001)

    MathSciNet  MATH  Google Scholar 

  8. Bosma, W., Cannon, J., Playoust, C.: The magma algebra system i: the user language. J. Symb. Comp. 24, 235–265. (1997) (See also http://magma.maths.usyd.edu.au/magma/). Accessed July 2021

  9. Bugeaud, Y.: On the greatest prime factor of \(ax^m-by^n\) II. Bull. Lond. Math. Soc. 32(6), 673–678 (2000)

    Article  MathSciNet  Google Scholar 

  10. Bugeaud, Y., Győry, K.: Bounds for the solutions of Thue–Mahler equations and norm form equations. Acta Arith. 74, 273–292 (1996)

    Article  MathSciNet  Google Scholar 

  11. Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Ann. Math. 163, 969–1018 (2006)

    Article  MathSciNet  Google Scholar 

  12. Carmichael, R.D.: On the numerical factors of the arithmetic forms \(\alpha ^n \pm \beta ^n\). Ann. Math. 15, 30–70 (1913)

    Article  MathSciNet  Google Scholar 

  13. Darmon, H., Merel, L.: Winding quotients and some variants of Ferma’s Last Theorem. J. Reine Angew. Math. 490, 81–100 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Deligne, P.: La conjecture de Weil I. Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974)

    Article  MathSciNet  Google Scholar 

  15. Dembner, S., Jain, V.: Hyperelliptic curves and newform coefficients. J. Number Theory 225, 214–239 (2021)

  16. Derickx, M., Van Hoeij, M., Zeng, Jinxiang: Computing Galois representations and equations for modular curves \(X_H(\ell )\), arXiv:1312.6819v2

  17. Gherga, A., von Känel, R., Matschke, B., Siksek, S.: Efficient resolution of Thue–Mahler equations. (to appear)

  18. Hanada, M., Madhukara, R.: Fourier coefficients of level \(1\) Hecke eigenforms. Acta Arith. (to appear)

  19. Kraus, A., Oesterlé, J.: Sur une question de B. Mazur. Math. Ann. 293, 259–275 (2002)

    Article  MathSciNet  Google Scholar 

  20. Lehmer, D.H.: The vanishing of Ramanujan’s function \(\tau (n)\). Duke Math. J. 14, 429–433 (1947)

    Article  MathSciNet  Google Scholar 

  21. Lenstra, H.W.: Algorithms in algebraic number theory. Bull. Am. Math. Soc. 26, 211–244 (1992)

    Article  MathSciNet  Google Scholar 

  22. van der Linden, F.J.: Class number computations of real abelian number fields. Math. Comput. 39, 693–707 (1982)

    Article  MathSciNet  Google Scholar 

  23. The LMFDB Collaboration, The L-functions and modular forms database (2021). http://www.lmfdb.org, Online; Accessed 7 July 2021

  24. Luca, F., Mabaso, S., Stanica, P.: On the prime factors of the iterates of the Ramanujan \(\tau \)-function. Proc. Edinburgh Math. Soc. Acta Arithmetica 63, 1031–1047 (2020)

  25. Luca, F., Shparlinski, I.E.: Arithmetic properties of the Ramanujan function. Proc. Indian Acad. Sci. (Math. Sci.) 116, 1–8 (2006)

    Article  MathSciNet  Google Scholar 

  26. Lygeros, N., Rozier, O.: A new solution to the equation \(\tau (p) \equiv 0 ~mod \;p\). J. Integer Seq. 13, Article 10.7.4 (2010)

  27. Lygeros, N., Rozier, O.: Odd prime values of the Ramanujan tau function. Ramanujan J. 32, 269–280 (2013)

    Article  MathSciNet  Google Scholar 

  28. Momose, F.: Isogenies of prime degree over number fields. Compos. Math. 97, 329–348 (1995)

    MathSciNet  MATH  Google Scholar 

  29. Mordell, L.J.: On Mr. Ramanujan’s empirical expansions of modular functions. Proc. Camb. Philos. Soc. 19, 117–124 (1917)

    MATH  Google Scholar 

  30. Ono, K., Taguchi, Y.: \(2\)-adic properties of certain modular forms and their applications to arithmetic functions. Int. J. Number Theory 1, 75–101 (2005)

    Article  MathSciNet  Google Scholar 

  31. Ram Murty, M., Kumar Murty, V.: Odd values of Fourier coefficients of certain modular forms. Int. J. Number Theory 3, 455–470 (2007)

    Article  MathSciNet  Google Scholar 

  32. Ram Murty, M., Kumar Murty, V., Shorey, T.N.: Odd values of the Ramanujan \(\tau \)-function. Bull. Soc. Math. France 115, 391–395 (1987)

    Article  MathSciNet  Google Scholar 

  33. Ramanujan, S.: On certain arithmetical functions. Trans. Camb. Philos. Soc. 22, 159–184 (1916)

    MATH  Google Scholar 

  34. Serre, J.-P.: Divisibilité de certaines fonctions arithmétiques. L’Ens. Math. 22(176), 227–260 (1974)

    MATH  Google Scholar 

  35. Siksek S: The modular approach to diophantine equations. In: Belabas, et al. (eds.) Explicit Methods in Number Theory: Rational Points and Diophantine Equations. Panoramas et synthèses, vol. 36. (2012)

  36. Smart, N.P.: The Algorithmic Resolution of Diophantine Equations. Cambridge University Press, Cambridg (1998)

    Book  Google Scholar 

  37. Stewart, C.L.: On divisors of Lucas and Lehmer numbers. Acta Math. 211, 291–314 (2013)

    Article  MathSciNet  Google Scholar 

  38. Swinnerton-Dyer, H.P.F.: On \(\ell \)-adic representations and congruences for coefficients of modular forms, page 1–55 of W. Kuyk and J.-P. Serre (eds.), Modular functions of one variable, III, Lecture Notes in Mathematics 350, (1973)

  39. Washington, L.C.: Introduction to Cyclotomic Fields. Springer, Berlin (1982)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Michael A. Bennett

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Adela Gherga & Samir Siksek

  3. Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Vandita Patel

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  1. Michael A. Bennett
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  2. Adela Gherga
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  4. Samir Siksek
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Correspondence to Michael A. Bennett.

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Communicated by Kannan Soundararajan.

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Michael A. Bennett is supported by NSERC. Adela Gherga and Samir Siksek are supported by an EPSRC Grant EP/S031537/1 “Moduli of elliptic curves and classical Diophantine problems”

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Bennett, M.A., Gherga, A., Patel, V. et al. Odd values of the Ramanujan tau function. Math. Ann. 382, 203–238 (2022). https://doi.org/10.1007/s00208-021-02241-3

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  • DOI: https://doi.org/10.1007/s00208-021-02241-3

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