Ono, Ken; Robins, Sinai; Wahl, Patrick T. On the representation of integers as sums of triangular numbers. (English) Zbl 0828.11057 Aequationes Math. 50, No. 1-2, 73-94 (1995). This survey article treats \(\delta_k (n)\), the number of representations of a positive integer \(n\) as the sum of \(k\) triangular numbers. Using the theory of modular forms the authors calculate \(\delta_k (n)\) explicitly for \(k = 2\), 3, 4, 6, 8, 10, 12 and 24. The results for \(k = 24\) reveal connections with 24-dimensional Leech lattices and with Ramanujan’s function \(\tau (n)\). It is also shown that the number of lattice points in a \(k\)-dimensional sphere of radius \(R\) centered at \(({1 \over 2}, {1 \over 2}, \ldots, {1 \over 2})\) is \(2^k\) times the sum \(\sum \delta_k (n)\) extended over all \(n \leq (4R^2 - k)/8\). Reviewer: T.M.Apostol (Pasadena) Cited in 1 ReviewCited in 18 Documents MSC: 11P82 Analytic theory of partitions 11F11 Holomorphic modular forms of integral weight 11P21 Lattice points in specified regions 11H06 Lattices and convex bodies (number-theoretic aspects) Keywords:representation of integers; sums of triangular numbers; number of lattice points in a \(k\)-dimensional sphere; survey; modular forms; 24-dimensional Leech lattices; Ramanujan’s function PDF BibTeX XML Cite \textit{K. Ono} et al., Aequationes Math. 50, No. 1--2, 73--94 (1995; Zbl 0828.11057) Full Text: DOI EuDML OpenURL Online Encyclopedia of Integer Sequences: Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n). Expansion of Product_{k>=1} (1 - x^k)^12. Fourier coefficients of E_{infinity,4}. Theta series of Leech lattice. Sum of divisors of 2*n + 1. Expansion of Jacobi theta constant theta_2^5 /32. Expansion of Jacobi theta constant theta_2^6 /(64q^(3/2)). Number of ways of writing n as the sum of 2 triangular numbers. Number of ordered ways of writing n as the sum of 3 triangular numbers. a(n) = 1 if n is a triangular number, otherwise 0. Expansion of Jacobi theta constant (theta_2/2)^12. Expansion of Jacobi theta constant (theta_2/2)^24. Glaisher’s chi_4(n). a(n) = Sum_{d|n, d==1 mod 4} d^4 - Sum_{d|n, d==3 mod 4} d^4. Sum {0<d|n, n/d odd} d^11. a(n) = 1 if n is an odd square, otherwise 0. Coefficients of the series expansion of q^(-1/4) pi_q. Number of ways of writing n as the sum of 7 triangular numbers. Number of ways of writing n as the sum of 9 triangular numbers. Number of ways of writing n as the sum of 10 triangular numbers from A000217. Number of ways of writing n as the sum of 11 triangular numbers. References: [1] Andrews, G.,Eureka! num = {\(\Delta\)} + {\(\Delta\)} + {\(\Delta\)}. J. Number Theory23 (1986), 285–293. · Zbl 0586.10007 [2] Andrews, G.,The theory of partitions. [Encyclopedia of Math., Vol. 2]. Addison-Wesley, Reading, MA, 1976. · Zbl 0371.10001 [3] Conway, J. andSloane, N.,Sphere packings, lattices and groups. Springer-Verlag, 1988. [4] Deligne, P.,Formes modulaires et representations l-adiques. In Seminaire Bourbaki [Lect. Notes in Math., No. 179], Springer-Verlag, Berlin, 1971. · Zbl 0206.49901 [5] Deligne, P. andSerre, J.-P.,Formes modulaires de poids 1. Ann. Scient. Ecole Norm. Sup. (4) 7 (1974). · Zbl 0321.10026 [6] Dickson, L.,Theory of numbers, Vol. III. Chelsea, New York, 1952. [7] Garvan, F., Kim, D. andStanton, D.,Cracks and t-cores. Invent. Math.101 (1990), 1–17. · Zbl 0721.11039 [8] Garvan, F.,Some congruence properties for partitions that are p-cores. Proc. London Math. Soc.66 (1993), 449–478. · Zbl 0788.11044 [9] Gordon, B. andRobins, S.,Lacunarity of Dedekind {\(\eta\)}-products. Glasgow Math. J.,37 (1995), 1–14. · Zbl 0817.11027 [10] Grosswald, E.,Representations of integers as sums of squares. Springer-Verlag, 1985. · Zbl 0574.10045 [11] Hida, H.,Elementary theory of L-functions and Eisenstein series. [London Math. Society Student Text, No. 26]. Cambridge Univ. Press, Cambridge, 1993. · Zbl 0942.11024 [12] Koblitz, N.,Introduction to elliptic curves and modular forms. Springer-Verlag, Berlin, 1984. · Zbl 0553.10019 [13] Kolberg, O.,Congruences for Ramanujan’s function {\(\tau\)}(n). [Årbok Univ. Bergen, Mat.-Natur. Ser. No. 1]. Univ., Bergen, 1962. · Zbl 0109.03101 [14] Legendre, A.,Traitée des fonctions elliptiques, Vol. 3, Paris, 1828. [15] Miyake T,Modular forms. Springer-Verlag, Berlin, 1989. [16] Ono, K.,Congruences on the Fourier coefficients of modular forms on Г0(N). Contemp. Math, to appear. · Zbl 0812.11029 [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. [18] Ono, K.,On the positivity of the number of t-core partitions. Acta Arithmetica66 (1994), 221–228. · Zbl 0793.11027 [19] Rankin, R.,Ramanujan’s unpublished work on congruences. [Lect. Notes in Math., No. 601]. Springer Verlag, Berlin, 1976. · Zbl 0359.10021 [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. · Zbl 0061.07207 [21] Robins, S.,Arithmetic properties of modular forms. Ph.D. Thesis, University of California, Los Angeles, 1991. [22] Robins, S.,Generalized Dedekind {\(\eta\)}-product. To appear in Contemp. Math. [23] Schoeneberg, B.,Elliptic modular functions–an introduction. Springer-Verlag, Berlin, 1970. · Zbl 0285.10016 [24] Serre, J. P.,Sur la lacunarite’ des puissances de {\(\eta\)}. Glasgow Math. J.27 (1985), 203–221. · Zbl 0583.10015 [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. [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. [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. [28] Shimura, G.,Introduction to the arithmetic theory of automorphic functions. [Publ. Math. Soc. of Japan, No. 11], Iwanami Shoten, Tokyo, 1971. · Zbl 0221.10029 [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. · Zbl 0267.10032 [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. · Zbl 0392.10030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.