The Ramanujan Journal

, Volume 18, Issue 1, pp 113–119 | Cite as

Simple upper bounds for partition functions



We tweak Siegel’s method to produce a rather simple proof of a new upper bound on the number of partitions of an integer into an exact number of parts. Our approach, which exploits the delightful dilogarithm function, extends easily to numerous other counting functions.


Partitions Dilogarithm 

Mathematics Subject Classification (2000)



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  1. 1.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998) MATHGoogle Scholar
  2. 2.
    Apostol, T.M.: Introduction to Analytic Number Theory, corrected 5th edn. Undergraduate Texts in Mathematics. Springer, New York (1998) Google Scholar
  3. 3.
    Beukers, F., Kolk, J.A.C., Calabi, E.: Sums of generalized harmonic series and volumes. Nieuw Arch. Wisk. 11(4), 217–224 (1993) MATHMathSciNetGoogle Scholar
  4. 4.
    Erdős, P., Lehner, J.: The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8, 335–345 (1941) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Gupta, H.: On an asymptotic formula in partitions. Proc. Indian Acad. Sci. A16, 101–102 (1942) Google Scholar
  6. 6.
    Knopp, M.I.: Modular Functions in Analytic Number Theory. AMS Chelsea, Providence (1993) MATHGoogle Scholar
  7. 7.
    van Lint, J.H.: Combinatorial Theory Seminar Eindhoven University of Technology. Lecture Notes in Mathematics, vol. 382. Springer, Berlin (1974) MATHGoogle Scholar
  8. 8.
    Nathanson, M.B.: Elementary Methods in Number Theory. Graduate Texts in Mathematics, vol. 195. Springer, New York (2000) MATHGoogle Scholar
  9. 9.
    Patterson, S.J.: An Introduction to the Theory of the Riemann Zeta-Function. Cambridge Studies in Advanced Mathematics, vol. 14. Cambridge University Press, Cambridge (1995) MATHGoogle Scholar
  10. 10.
    Pribitkin, W.: Revisiting Rademacher’s formula for the partition function p(n). Ramanujan J. 4, 455–467 (2000) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rademacher, H.: A convergent series for the partition function p(n). Proc. Nat. Acad. Sci. U.S.A. 23, 78–84 (1937) CrossRefGoogle Scholar
  12. 12.
    Rademacher, H.: On the partition function p(n). Proc. Lond. Math. Soc. 43(2), 241–254 (1937) MATHGoogle Scholar
  13. 13.
    Rademacher, H.: Topics in Analytic Number Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 169. Springer, New York (1973) MATHGoogle Scholar
  14. 14.
    Riordan, J.: An Introduction to Combinatorial Analysis. Princeton University Press, Princeton (1978) Google Scholar

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Mathematics, College of Staten IslandCUNYStaten IslandUSA

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