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. If n ≥0 is a non-negative integer, then the nth triangular number is T n = n(n + 1)/2. Let k be a positive integer. We denote by δ k (n) the number of representations of n as a sum of k triangular numbers. Here we use the theory of modular forms to calculate δ k (n). The case where k = 24 is particularly interesting. It turns out that, if n ≥3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 212(212 − 1)δ 24(n − 3). Furthermore the formula for δ 24(n) involves the Ramanujan τ(n)-function. As a consequence, we get elementary congruences for τ(n). In a similar vein, when p is a prime, we demonstrate δ 24 (p k − 3) as a Dirichlet convolution of σ 11(n) and τ (n). It is also of interest to know that this study produces formulas for the number of lattice points inside k-dimensional spheres.
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References
Andrews, G., Eureka! num = Δ + Δ + Δ. J. Number Theory 23 (1986), 285–293.
Andrews, G., The theory of partitions. [Encyclopedia of Math., Vol. 2]. Addison-Wesley, Reading, MA, 1976.
Conway, J. and sloane, N., Sphere packings, lattices and groups. Springer-Verlag, 1988.
Deligne, P., Formes modulaires et representations 1-adiques. In Seminaire Bourbaki [Lett. Notes in Math., No. 179], Springer-Verlag, Berlin, 1971.
Deligne, P. and Serre, J.-P., Formes modulaires de poids 1. Ann. Scient. Ecole Norm. Sup. (4) 7 (1974).
Dickson, L., Theory of numbers, Vol. III. Chelsea, New York, 1952.
Garvan, F., Kim, D. and Stanton, D., Cracks and t-cores. Invent. Math. 101 (1990), 1–17.
Garvan, F., Some congruence properties for partitions that are p-cores. Proc. London Math. Soc. 66 (1993),449–478.
Gordon, B. and Robins, S., Lacunarity of Dedekind q-products. Glasgow Math. J., 37 (1995), 1–14.
Grosswald, E., Representations of integers as sums of squares. Springer-Verlag, 1985.
Hida, H., Elementary theory of L-functions and Eisenstein series. [London Math. Society Student Text, No. 26]. Cambridge Univ. Press, Cambridge, 1993.
Koblitz, N., Introduction to elliptic curves and modular forms. Springer-Verlag, Berlin, 1984.
Kolberg, O., Congruences for Ramanujan’s function r(n). [Arbok Univ. Bergen, Mat.-Natur. Ser. No. 1]. Univ., Bergen, 1962.
Legendre, A.,Traitée des fonctions elliptiques Vol. 3Paris, 1828.
Miyake TModular forms. Springer-Verlag, Berlin, 1989.
Ono, k., Congruences on the Fourier coefficients of modular forms on Г 0 (N). Contemp. Math, to appear.
Ono, k., Congruences on the Fourier coefficients of modular forms on Г 0 (N) with number-theoretic applications. Ph.D. Thesis, University of California, Los Angeles, 1993.
Ono, k., On the positivity of the number of t-core partitions. Acta Arithmetica 66 (1994), 221–228.
Rankin, R., Ramanujan’s unpublished work on congruences. [Lect. Notes in Math., No. 601]. Springer Verlag, Berlin, 1976.
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.
Robins, S., Arithmetic properties of modular forms. Ph.D. Thesis, University of California, Los Angeles, 1991.
Robins, S., Generalized Dedekind q-product. To appear in Contemp. Math.
Schoeneberg, B., Elliptic modular functions —an introduction. Springer-Verlag, Berlin, 1970.
Serre, J. P., Sur la lacunarite’ des puissances de rl. Glasgow Math. J. 27 (1985), 203–221.
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.
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.
Serre, J.-P. and Stark, H., Modular forms of weight z. In modular functions of one variable, Vol. VI. [Lect. Notes in Math., No. 627]. Springer Verlag, Berlin, 1971, pp. 27–67.
Shimura, G., Introduction to the arithmetic theory of automorphic functions. [Publ. Math. Soc. of Japan, No. 11], Iwanami Shoten, Tokyo, 1971.
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.
Swinnerton-Dyer, H. P. F., On 1-adic representations and congruences for coefficients of modular forms II. [Lect. Notes in Math., No. 601]. Springer Verlag, Berlin, 1976.
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Ono, K., Robins, S., Wahl, P.T. (1995). On the representation of integers as sums of triangular numbers. In: Aczél, J. (eds) Aggregating clones, colors, equations, iterates, numbers, and tiles. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9096-0_6
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DOI: https://doi.org/10.1007/978-3-0348-9096-0_6
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