Advertisement

Jacobi’s Four and Eight Squares Theorems and Partitions into Distinct Parts

  • Cristina Ballantine
  • Mircea MercaEmail author
Article
  • 46 Downloads

Abstract

We consider the function \(r_s(n)\) which gives the number of ways to write n as the sum of s squares. Since the generating functions for \(r_4(n)\) and \(r_8(n)\) are Lambert series, we use Merca’s factorization theorem for Lambert series to establish relationships between these functions and partitions into distinct parts. We also obtain convolutions involving overpartition functions as well as pentagonal recurrence formulas for \(r_4(n)\) and \(r_8(n)\). These results lead to new connections between divisors and partitions.

Keywords

Sum of squares Sum of divisors Partitions into distinct parts 

Mathematics Subject Classification

05A17 05A19 11E25 11A25 11P81 

Notes

References

  1. 1.
    Andrews, G.E., Ekhad, S.B., Zeilberger, D.: A short proof of Jacobi’s formula for the number of representations of an integer as a sum of four squares. Am. Math. Mon. 100(3), 274–276 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions. Addison-Wesley Publishing, New York (1976)zbMATHGoogle Scholar
  3. 3.
    Andrews, G.E.: Singular overpartitions. Int. J. Number Theory 11(5), 1523–1533 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ballantine, C., Merca, M.: New convolutions for the number of divisors. J. Number Theory 170, 17–34 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley, New York (1987)zbMATHGoogle Scholar
  6. 6.
    Chan, H.H., Krattenthaler, C.: Recent progress in the study of representations of integers as sums of squares. Bull. Lond. Math. Soc. 37(6), 818–826 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn., vol. 290 of Grundlehren der Mathematischen Wissenschaften. Springer, New York (1999)CrossRefGoogle Scholar
  8. 8.
    Corteel, S., Lovejoy, J.: Overpartitions. Trans. Am. Math. Soc. 356, 1623–1635 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Glasser, M.L., Zucker, I.J.: Lattice sums. In: Eyring, H., Henderson, D. (eds.) Theoretical Chemistry: Advances and Perspectives, vol. 5, pp. 67–139. Academic Press, New York (1980)CrossRefGoogle Scholar
  10. 10.
    Grosswald, E.: Representations of Integers as Sums of an Even Number of Squares. Springer, New York (1985)CrossRefGoogle Scholar
  11. 11.
    Hirschhorn, M.D.: A simple proof of Jacobi’s four-square theorem. Proc. Am. Math. Soc. 101(3), 436–438 (1987)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hirschhorn, M.D., Sellers, J.A.: On a problem of Lehmer on partitions into squares. Ramanujan J. 8(3), 279–287 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hirschhorn, M.D., Sellers, J.A.: Arithmetic relations for overpartitions. J. Comb. Math. Comb. Comp. 53, 65–73 (2005)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hirschhorn, M.D., Sellers, J.A.: Arithmetic properties of overpartitions into odd parts. Ann. Comb. 10(3), 353–367 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kim, B.: A short note on the overpartition function. Discrete Math. 309, 2528–2532 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lehmer, D.H.: On the partition of numbers into squares. Am. Math. Mon. 55, 476–481 (1948)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lovejoy, J.: Gordon’s theorem for overpartitions. J. Comb. Theory Ser. A 103, 393–401 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lovejoy, J.: Overpartition theorems of the Rogers–Ramaujan type. J. Lond. Math. Soc. 69, 562–574 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lovejoy, J.: Overpartitions and real quadratic fields. J. Number Theory 106, 178–186 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lubotzky, A., Phillips, R.: P. Sarnak Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mahlburg, K.: The overpartition function modulo small powers of 2. Discrete Math. 286, 263–267 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Merca, M.: A new look on the generating function for the number of divisors. J. Number Theory 149, 57–69 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Merca, M.: Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer. J. Number Theory 160, 60–75 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Merca, M.: New relations for the number of partitions with distinct even parts. J. Number Theory 176, 1–12 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Merca, M.: From a Rogers’s identity to overpartitions. Period. Math. Hungar. 75, 172–179 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Merca, M.: The Lambert series factorization theorem. Ramanujan J. 44, 417–435 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Milne, S.C.: Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions. Ramanujan J. 6, 7–149 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rademacher, H.: Topics in Analytic Number Theory, Vol. 169 of Grundlehren Math. Wiss. Springer, New York (1973)CrossRefGoogle Scholar
  29. 29.
    Rankin, R.A.: On the representations of a number as a sum of squares and certain related identities. Proc. Camb. Philos. Soc. 41, 1–11 (1945)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Rankin, R.A.: On the representation of a number as the sum of any number of squares, and in particular of twenty. Acta Arith. 7, 399–407 (1962)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rankin, R.A.: Sums of squares and cusp forms. Am. J. Math. 87, 857–860 (1965)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. http://oeis.org (2018)
  33. 33.
    Taussky, O.: Sums of squares. Am. Math. Mon. 77, 805–830 (1970)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Williams, K.S.: An arithmetic proof of Jacobi’s eight squares theorem. Far East J. Math. 3(6), 1001–1005 (2001)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Williams, K.S.: The parents of Jacobi’s four squares theorem are unique. Am. Math. Mon. 120, 329–345 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceCollege of The Holy CrossWorcesterUSA
  2. 2.Academy of Romanian ScientistsBucharestRomania

Personalised recommendations