The asymptotic distribution of Andrews’ smallest parts function

Abstract

In this paper, we use methods from the spectral theory of automorphic forms to give an asymptotic formula with a power saving error term for Andrews’ smallest parts function spt(n). We use this formula to deduce an asymptotic formula with a power saving error term for the number of 2-marked Durfee symbols associated to partitions of n. Our method requires that we count the number of Heegner points of discriminant −D < 0 and level N inside an “expanding” rectangle contained in a fundamental domain for \({\Gamma_0(N)}\).

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References

  1. 1.

    S. Ahlgren and N. Andersen, Algebraic and transcendental formulas for the smallest parts function, Preprint available at: arXiv:1504.02500

  2. 2.

    Ahlgren S., Bringmann K., Lovejoy J.: \({\ell}\)-adic properties of smallest parts functions. Adv. Math. 228, 629–645 (2011)

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Andrews G. E.: Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks. Invent. Math. 169, 37–73 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Andrews G. E.: The number of smallest parts in the partitions of n, J. Reine Angew. Math. 624, 133–142 (2008)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Atkin A. O. L., Garvan F.: Relations between the ranks and cranks of partitions. Ramanujan J. 7, 343–366 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    Bringmann K.: On the explicit construction of higher deformations of partition statistics. Duke Math. J. 144, 195–233 (2008)

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Bringmann K., Ono K.: An arithmetic formula for the partition function. Proc. Amer. Math. Soc. 135, 3507–3514 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    Bruinier J. H., Jenkins P., Ono K.: Hilbert class polynomials and traces of singular moduli. Math. Ann. 334, 373–393 (2006)

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Bruinier J. H., Ono K.: Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms. Adv. Math. 246, 198–219 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Duke W.: Modular functions and the uniform distribution of CM points. Math. Ann. 334, 241–252 (2006)

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

    Folsom A., Masri R.: Equidistribution of Heegner points and the partition function. Math. Ann. 348, 289–317 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Folsom A., Ono K.: The spt-function of Andrews. Proc. Natl. Acad. Sci. USA 105, 20152–20156 (2008)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Garvan F.: Congruences for Andrews’ smallest parts partition function and new congruences for Dyson’s rank. Int. J. Number Theory 6, 1–29 (2010)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gross B., Zagier D.: Heegner points and derivatives of L-series. Invent. Math. 84, 225–320 (1986)

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    H. Iwaniec, Introduction to the spectral theory of automorphic forms, Biblioteca de la Revista Matemática Iberoamericana. Revista Matematica Iberoamericana, Madrid, 1995. xiv+247 pp.

  16. 16.

    H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, 17. American Mathematical Society, Providence, RI, 1997. xii+259 pp.

  17. 17.

    R. Masri, Fourier coefficients of harmonic weak Maass forms and the partition function, Amer. J. Math. 137, 1061–1097 (2015)

  18. 18.

    R. Masri, Singular moduli and the distribution of partition ranks modulo 2, Mathematical Proceedings of the Cambridge Philosophical Society, to appear.

  19. 19.

    Ono K.: Congruences for the Andrews spt function. Proc. Natl. Acad. Sci. USA 108, 473–476 (2011)

    MATH  MathSciNet  Article  Google Scholar 

  20. 20.

    M. P. Young, Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets, Journal of the European Mathematical Society, to appear. Preprint available at: arXiv:1405.5457v2

  21. 21.

    D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), 211–244, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002.

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Correspondence to Riad Masri.

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Banks, J., Barquero-Sanchez, A., Masri, R. et al. The asymptotic distribution of Andrews’ smallest parts function. Arch. Math. 105, 539–555 (2015). https://doi.org/10.1007/s00013-015-0831-9

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Mathematics Subject Classification

  • 11M41

Keywords

  • Durfee symbol
  • Partition
  • Smallest parts function