Advertisement

Asymptotic formulas related to the \(M_2\)-rank of partitions without repeated odd parts

  • Chris Jennings-ShafferEmail author
  • Dillon Reihill
Article
  • 2 Downloads

Abstract

We give asymptotic expansions for the moments of the \(M_2\)-rank generating function and for the \(M_2\)-rank generating function at roots of unity. For this we apply the Hardy–Ramanujan circle method extended to mock modular forms. Our formulas for the \(M_2\)-rank at roots of unity lead to asymptotics for certain combinations of N2(rmn) (the number of partitions without repeated odd parts of n with \(M_2\)-rank congruent to r modulo m). This allows us to deduce inequalities among certain combinations of N2(rmn). In particular, we resolve a few conjectured inequalities of Mao.

Keywords

Integer partitions Partition ranks Rank differences Asymptotics Circle method Rank inequalities \(M_2\)-rank Harmonic Maass forms Modular forms Mock modular forms Mock theta functions 

Mathematics Subject Classification

Primary 11P82 11P81 

Notes

Acknowledgements

The authors thank Kathrin Bringmann for suggesting this project and for useful comments and discussions.

References

  1. 1.
    Alwaise, E., Iannuzzi, E., Swisher, H.: A proof of some conjectures of Mao on partition rank inequalities. Ramanujan J. 43(3), 633–648 (2017)CrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, G.E.: On the theorems of Watson and Dragonette for Ramanujan’s mock theta functions. Am. J. Math. 88, 454–490 (1966)CrossRefGoogle Scholar
  3. 3.
    Andrews, G.E.: The Theory of Partitions. Encyclopedia of Mathematics and Its Applications, vol. 2. Addison-Wesley Publishing Co., Reading (1976)Google Scholar
  4. 4.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part I. Springer, New York (2005)zbMATHGoogle Scholar
  5. 5.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part III. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  6. 6.
    Atkin, A.O.L., Swinnerton-Dyer, P.: Some properties of partitions. Proc. London Math. Soc. 3(4), 84–106 (1954)CrossRefzbMATHGoogle Scholar
  7. 7.
    Barman, R., Sachdeva, A.P.S.: Proof of a limited version of Mao’s partition rank inequality using a theta function identity. Res. Number Theory 2, 6 (2016)CrossRefzbMATHGoogle Scholar
  8. 8.
    Berkovich, A., Garvan, F.G.: Some observations on Dyson’s new symmetries of partitions. J. Combin. Theory Ser. A 100(1), 61–93 (2002)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bringmann, K.: Asymptotics for rank partition functions. Trans. Am. Math. Soc. 361(7), 3483–3500 (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bringmann, K., Folsom, A.: On the asymptotic behavior of Kac-Wakimoto characters. Proc. Am. Math. Soc. 141(5), 1567–1576 (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass Forms and Mock Modular Forms: Theory and Applications. American Mathematical Society Colloquium Publications, vol. 64. American Mathematical Society, Providence, RI (2017)CrossRefzbMATHGoogle Scholar
  12. 12.
    Bringmann, K., Mahlburg, K.: Asymptotic formulas for coefficients of Kac-Wakimoto characters. Math. Proc. Cambridge Philos. Soc. 155(1), 51–72 (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Bringmann, K., Mahlburg, K., Rhoades, R.C.: Taylor coefficients of mock-Jacobi forms and moments of partition statistics. Math. Proc. Cambridge Philos. Soc. 157(2), 231–251 (2014)CrossRefzbMATHGoogle Scholar
  14. 14.
    Bringmann, K., Ono, K.: The \(f(q)\) mock theta function conjecture and partition ranks. Invent. Math. 165(2), 243–266 (2006)CrossRefzbMATHGoogle Scholar
  15. 15.
    Bringmann, K., Ono, K.: Dyson’s ranks and Maass forms. Ann. Math. (2) 171(1), 419–449 (2010)CrossRefzbMATHGoogle Scholar
  16. 16.
    Dragonette, L.A.: Some asymptotic formulae for the mock theta series of Ramanujan. Trans. Am. Math. Soc. 72, 474–500 (1952)CrossRefzbMATHGoogle Scholar
  17. 17.
    Dyson, F.J.: Some guesses in the theory of partitions. Eureka 8, 10–15 (1944)Google Scholar
  18. 18.
    Garvan, F.G., Jennings-Shaffer, C.: The spt-crank for overpartitions. Acta Arith. 166(2), 141–188 (2014)CrossRefzbMATHGoogle Scholar
  19. 19.
    Gordon, B., McIntosh, R.J.: A survey of classical mock theta functions. In: Alladi, K. (ed.) Partitions, \(q\)-Series, and Modular Forms. Developments in Mathematics, vol. 23, pp. 95–144. Springer, New York (2012)CrossRefGoogle Scholar
  20. 20.
    Hardy, G.H., Ramanujan, S.: Asymptotic Formulaae in Combinatory Analysis. Proc. London Math. Soc. 2(17), 75–115 (1918)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hickerson, D.R., Mortenson, E.T.: Hecke-type double sums, Appell-Lerch sums, and mock theta functions. I. Proc. Lond. Math. Soc. (3) 109(2), 382–422 (2014)CrossRefzbMATHGoogle Scholar
  22. 22.
    Knopp, M .I.: Modular Functions in Analytic Number Theory. Markham Publishing Co., Chicago, Ill (1970)zbMATHGoogle Scholar
  23. 23.
    Lovejoy, J., Osburn, R.: \(M_2\)-rank differences for partitions without repeated odd parts. J. Théor. Nombres Bordeaux 21(2), 313–334 (2009)CrossRefzbMATHGoogle Scholar
  24. 24.
    Mao, R.: Asymptotics for rank moments of overpartitions. Int. J. Number Theory 10(8), 2011–2036 (2014)CrossRefzbMATHGoogle Scholar
  25. 25.
    Mao, R.: The \(M_2\)-rank of partitions without repeated odd parts modulo \(6\) and \(10\). Ramanujan J. 37(2), 391–419 (2015)CrossRefzbMATHGoogle Scholar
  26. 26.
    Mao, R.: Asymptotic formulas for \(M_2\)-ranks of partitions without repeated odd parts. J. Number Theory 166, 324–343 (2016)CrossRefzbMATHGoogle Scholar
  27. 27.
    McIntosh, R.J.: Second order mock theta functions. Canad. Math. Bull. 50(2), 284–290 (2007)CrossRefzbMATHGoogle Scholar
  28. 28.
    Rademacher, H.: On the partition function p(n). Proc. London Math. Soc. (2) 43(4), 241–254 (1937)zbMATHGoogle Scholar
  29. 29.
    Shimura, G.: On modular forms of half integral weight. Ann. Math. 2(97), 440–481 (1973)CrossRefzbMATHGoogle Scholar
  30. 30.
    Waldherr, M.: Asymptotics for moments of higher ranks. Int. J. Number Theory 09(03), 675–712 (2013)CrossRefzbMATHGoogle Scholar
  31. 31.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press/The Macmillan Company, Cambridge, England/New York (1944)zbMATHGoogle Scholar
  32. 32.
    Zagier, D.: Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann). Astérisque, (326):Exp. No. 986, vii–viii, 143–164 (2010), 2009. Séminaire Bourbaki. Vol. 2007/2008Google Scholar
  33. 33.
    Zwegers, S.P.: Mock theta functions. PhD thesis, Universiteit Utrecht (2002)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of CologneCologneGermany

Personalised recommendations