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The f(q) mock theta function conjecture and partition ranks

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Abstract

In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for N e (n) (resp. N o (n)), the number of partitions of n with even (resp. odd) rank. Thanks to Rademacher’s celebrated formula for the partition function, this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f(q), a problem with its own long history dating to Ramanujan’s last letter to Hardy. Little was known about this problem until Dragonette in 1952 obtained asymptotic results. In 1966, G.E. Andrews refined Dragonette’s results, and conjectured an exact formula for the coefficients of f(q). By constructing a weak Maass-Poincaré series whose “holomorphic part” is q -1 f(q 24), we prove the Andrews-Dragonette conjecture, and as a consequence obtain the desired formulas for N e (n) and N o (n).

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References

  1. Abramowitz, M., Stegun, I.: Pocketbook of Mathematical Functions. Frankfurt: Harri Deutsch 1984

  2. Andrews, G.E.: On the theorems of Watson and Dragonette for Ramanujan’s mock theta functions. Am. J. Math. 88, 454–490 (1966)

    Google Scholar 

  3. Andrews, G.E.: Mock theta functions, Theta functions – Bowdoin 1987, part 2 (Brunswick, ME 1987). Proc. Sympos. Pure Math. vol. 49, part 2, pp. 283–297. Providence, RI: Am. Math. Soc. 1989

  4. Andrews, G.E.: Partitions: At the interface of q-series and modular forms. Rankin Memorial Issues, Ramanujan J. 7, 385–400 (2003)

    MATH  Google Scholar 

  5. Andrews, G.E.: Partitions with short sequences and mock theta functions. Proc. Natl. Acad. Sci. USA 102, 4666–4671 (2005)

    Article  MathSciNet  Google Scholar 

  6. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge: Cambridge Univ. Press 1999

  7. Andrews, G.E., Dyson, F., Hickerson, D.: Partitions and indefinite quadratic forms. Invent. Math. 91, 391–407 (1988)

    Article  MathSciNet  Google Scholar 

  8. Andrews, G.E., Garvan, F.: Dyson’s crank of a partition. Bull. Am. Math. Soc., New Ser. 18, 167–171 (1988)

    Article  MATH  Google Scholar 

  9. Atkin, A.O.L., Swinnerton-Dyer, H.P.F.: Some properties of partitions. Proc. Lond. Math. Soc. 66, 84–106 (1954)

    MathSciNet  Google Scholar 

  10. Bruinier, J.H.: Borcherds Products on O(2,ℓ) and Chern Classes of Heegner Divisors. Lect. Notes, vol. 1780. Berlin: Springer 2002

  11. Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125, 45–90 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Choi, Y.-S.: Tenth order mock theta functions in Ramanujan’s lost notebook. Invent. Math. 136, 497–569 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cohen, H.: q-identities for Maass waveforms. Invent. Math. 91, 409–422 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cohen, H.: Corrigendum to “q-identities for Maass waveforms”. Invent. Math. 120, 579–580 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Cohen, H., Oesterlé, J.: Dimensions des espaces des formes modulaires. Lect. Notes, vol. 627, pp. 69–78. Springer 1977

  16. Dragonette, L.: Some asymptotic formulae for the mock theta series of Ramanujan. Trans. Am. Math. Soc. 72, 474–500 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dyson, F.: Some guesses in the theory of partitions. Eureka (Cambridge) 8, 10–15 (1944)

    Google Scholar 

  18. Hejhal, D.A.: The Selberg Trace Formula for PSL2(ℝ), vol. II. Lect. Notes Math., vol. 1001. Springer 1983

  19. Hickerson, D.: On the seventh order mock theta functions. Invent. Math. 94, 661–677 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hooley, C.: On the number of divisors of a quadratic polynomial. Acta Math. 110, 97–114 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mahlburg, K.: Partition congruences and the Andrews-Garvan-Dyson crank. Proc. Natl. Acad. Sci. USA 102, 15373–15376 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rademacher, H.: Topics in Analytic Number Theory, Die Grundlehren der mathematischen Wissenschaften, Band 169. New York, Heidelberg: Springer 1973

  23. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa 1988

  24. Serre, J.-P., Stark, H.: Modular forms of weight 1/2. Lect. Notes, vol. 627, pp. 27–67. Springer 1977

  25. Shimura, G.: On modular forms of half integral weight. Ann. Math. 97, 440–481 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  26. Watson, G.N.: The final problem: An account of the mock theta functions. J. Lond. Math. Soc. 2, 55–80 (1936)

    Google Scholar 

  27. Zwegers, S.P.: Mock ϑ-functions and real analytic modular forms. q-Series with Applications to Combinatorics, Number Theory, and Physics, ed. by B.C. Berndt, K. Ono. Contemp. Math. 291, 269–277 (2001)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, Wisconsin, 53706, USA

    Kathrin Bringmann & Ken Ono

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  1. Kathrin Bringmann
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  2. Ken Ono
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Correspondence to Kathrin Bringmann or Ken Ono.

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Mathematics Subject Classification (2000)

11P82, 05A17

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Bringmann, K., Ono, K. The f(q) mock theta function conjecture and partition ranks. Invent. math. 165, 243–266 (2006). https://doi.org/10.1007/s00222-005-0493-5

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  • DOI: https://doi.org/10.1007/s00222-005-0493-5

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