aequationes mathematicae

, Volume 50, Issue 1–2, pp 73–94

# On the representation of integers as sums of triangular numbers

• Ken Ono
• Sinai Robins
• Patrick T. Wahl
Survey Papers

## Summary

In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn ≥ 0 is a non-negative integer, then thenth triangular number isT n =n(n + 1)/2. Letk be a positive integer. We denote byδ k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculateδ k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn ≥ 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 212(212 − 1)δ24(n − 3). Furthermore the formula forδ24(n) involves the Ramanujanτ(n)-function. As a consequence, we get elementary congruences forτ(n). In a similar vein, whenp is a prime, we demonstrateδ24(p k − 3) as a Dirichlet convolution ofσ11(n) andτ(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.

## AMS (1991) subject classification

Primary 11F11, 11F12, 11F37 Secondary 11P81

## References

1. [1]
Andrews, G.,Eureka! num = Δ + Δ + Δ. J. Number Theory23 (1986), 285–293.Google Scholar
2. [2]
3. [3]
Conway, J. andSloane, N.,Sphere packings, lattices and groups. Springer-Verlag, 1988.Google Scholar
4. [4]
Deligne, P.,Formes modulaires et representations l-adiques. In Seminaire Bourbaki [Lect. Notes in Math., No. 179], Springer-Verlag, Berlin, 1971.Google Scholar
5. [5]
Deligne, P. andSerre, J.-P.,Formes modulaires de poids 1. Ann. Scient. Ecole Norm. Sup. (4) 7 (1974).Google Scholar
6. [6]
Dickson, L.,Theory of numbers, Vol. III. Chelsea, New York, 1952.Google Scholar
7. [7]
Garvan, F., Kim, D. andStanton, D.,Cracks and t-cores. Invent. Math.101 (1990), 1–17.Google Scholar
8. [8]
Garvan, F.,Some congruence properties for partitions that are p-cores. Proc. London Math. Soc.66 (1993), 449–478.Google Scholar
9. [9]
Gordon, B. andRobins, S.,Lacunarity of Dedekind η-products. Glasgow Math. J.,37 (1995), 1–14.Google Scholar
10. [10]
Grosswald, E.,Representations of integers as sums of squares. Springer-Verlag, 1985.Google Scholar
11. [11]
Hida, H.,Elementary theory of L-functions and Eisenstein series. [London Math. Society Student Text, No. 26]. Cambridge Univ. Press, Cambridge, 1993.Google Scholar
12. [12]
Koblitz, N.,Introduction to elliptic curves and modular forms. Springer-Verlag, Berlin, 1984.Google Scholar
13. [13]
Kolberg, O.,Congruences for Ramanujan's function τ(n). [Årbok Univ. Bergen, Mat.-Natur. Ser. No. 1]. Univ., Bergen, 1962.Google Scholar
14. [14]
Legendre, A.,Traitée des fonctions elliptiques, Vol. 3, Paris, 1828.Google Scholar
15. [15]
Miyake T,Modular forms. Springer-Verlag, Berlin, 1989.Google Scholar
16. [16]
Ono, K.,Congruences on the Fourier coefficients of modular forms on Г0(N). Contemp. Math, to appear.Google Scholar
17. [17]
Ono, K.,Congruences on the Fourier coefficients of modular forms on Г0(N)with number-theoretic applications. Ph.D. Thesis, University of California, Los Angeles, 1993.Google Scholar
18. [18]
Ono, K.,On the positivity of the number of t-core partitions. Acta Arithmetica66 (1994), 221–228.Google Scholar
19. [19]
Rankin, R.,Ramanujan's unpublished work on congruences. [Lect. Notes in Math., No. 601]. Springer Verlag, Berlin, 1976.Google Scholar
20. [20]
Rankin, R. A.,On the representations of a number as a sum of squares and certain related identities. Proc. Cambridge Phil. Soc.41 (1945), 1–11.Google Scholar
21. [21]
Robins, S.,Arithmetic properties of modular forms. Ph.D. Thesis, University of California, Los Angeles, 1991.Google Scholar
22. [22]
Robins, S.,Generalized Dedekind η-product. To appear in Contemp. Math.Google Scholar
23. [23]
Schoeneberg, B.,Elliptic modular functions—an introduction. Springer-Verlag, Berlin, 1970.Google Scholar
24. [24]
Serre, J. P.,Sur la lacunarite' des puissances de η. Glasgow Math. J.27 (1985), 203–221.Google Scholar
25. [25]
Serre, J. P.,Quelques applications du theorme de densite de Chebotarev. [Publ. Math. I.H.E.S., No. 54]. Inst. Hautes Etudes Sci. Pub., I.H.E.S., Paris, 1981.Google Scholar
26. [26]
Serre, J. P.,Congruences et formes modulaires (d'apres H.P.F. Swinnerton-Dyer). In Seminaire Bourbaki, 24e anneé (1971/1972), Exp. No. 416. [Lect. Notes in Math., No. 317]. Springer Verlag, Berlin, 1973, pp. 319–338.Google Scholar
27. [27]
Serre, J.-P. andStark, H.,Modular forms of weight 1/2. In modular functions of one variable, Vol. VI. [Lect. Notes in Math., No. 627]. Springer Verlag, Berlin, 1971, pp. 27–67.Google Scholar
28. [28]
Shimura, G.,Introduction to the arithmetic theory of automorphic functions. [Publ. Math. Soc. of Japan, No. 11], Iwanami Shoten, Tokyo, 1971.Google Scholar
29. [29]
Swinnerton-Dyer, H. P. F.,On l-adic representations and congruences for coefficients of modular forms. [Lect. Notes in Math., No. 350]. Springer Verlag, Berlin, 1973.Google Scholar
30. [30]
Swinnerton-Dyer, H. P. F.,On l-adic representations and congruences for coefficients of modular forms II. [Lect. Notes in Math., No. 601]. Springer Verlag, Berlin, 1976.Google Scholar

## Authors and Affiliations

• Ken Ono
• 1
• Sinai Robins
• 2
• Patrick T. Wahl
• 3
1. 1.Department of MathematicsThe University of GeorgiaAthensUSA
2. 2.Department of Mathematical SciencesUniversity of Northern ColoradoGreeleyUSA
3. 3.Department of MathematicsUniversity of ColoradoBoulderUSA