that, for all n≥0
p{3,3}(3n+ 1) ≡p{3,3}(3n+ 2)≡0 (mod 3), and
p{3,3}(3n)≡
(−1)k+` (mod 3), if n=k(3k−1)/2 +`(3`−1)/2;
0 (mod 3), otherwise.
They also found some congruences for p{3,3}(n) modulo 4 and 9. They further conjectured four Ramanujan-like congruences modulo 5 satisfied by p{3,3}(n).
Conjecture 8.1. [17, Conjecture 5.1] For all n≥0,
p{3,3}(15n+ 6)≡0 (mod 5), (8.2) p{3,3}(25n+ 6)≡0 (mod 5), (8.3) p{3,3}(25n+ 16)≡0 (mod 5), (8.4) p{3,3}(25n+ 21)≡0 (mod 5). (8.5)
8.2 Proof of Conjecture 8.1
In this section we confirm that Conjecture8.1is true using the theory of modular forms.
Theorem 8.2. Conjecture 8.1 is true.
Proof. From (8.1), we have
∞
X
n=0
p{3,3}(n)qn = (q3;q3)3∞ (q;q)3∞ .
112 3-Regular Partitions in Three Colors
∆∗ and Pm,r(t) = {6}. By Lemma 7.4, we know that
1 0 δ 1
:δ|15
forms a
complete set of double coset representatives of Γ0(N)\Γ/Γ∞. Let γδ =
1 0 δ 1
. Let
r0 = (30,0,0,0)∈ R(15) and we use Sage to verify that pm,r(γδ) +p∗r0(γδ) ≥ 0 for each δ|N. Using Lemma 7.3 we have
ν := 1 24
X
δ|M
rδ+X
δ|N
rδ0
[Γ : Γ0(N)]−X
δ|N
δrδ0
− 1 24m
X
δ|M
δrδ− tmin m
= 1
24{(0 + 30) 24−30} − 1
24·15(−3 + 9)− 6 15 = 85
3.
Thereforebνcis equal to 28. Using Sagewe verify thatp{3,3}(15n+ 6)≡0 (mod 5) for all 0≤n ≤28. By Lemma7.3 we conclude that p{3,3}(15n+ 6)≡0 (mod 5) for alln≥0. This completes the proof of (8.2). To prove (8.3), we take (m, M, N, r, t) = (25,3,15,(−3,3),6). It is easy to verify that (m, M, N, r, t)∈∆∗ andPm,r(t) = {6}.
We compute that the upper bound in Lemma 7.3 is bνc = 47. Following similar steps as shown before, we find that p{3,3}(25n+ 6)≡0 (mod 5) for all n ≥0.
We now prove (8.4) and (8.5). We take (m, M, N, r, t) = (25,3,15,(−3,3),16).
It is easy to verify that (m, M, N, r, t) ∈ ∆∗ and Pm,r(t) = {16,21}. Here we also check that tmin = 16. By Lemma 7.4, we know that
1 0 δ 1
:δ|15
forms a
complete set of double coset representatives of Γ0(N)\Γ/Γ∞. Let γδ =
1 0 δ 1
. Let
r0 = (50,0,0,0)∈ R(15) and we use Sage to verify that pm,r(γδ) +p∗r0(γδ) ≥ 0 for eachδ|N. We compute that the upper bound in Lemma7.3isbνc= 47. UsingSage we verify that p{3,3}(25n+t0)≡ 0 (mod 5) for all t0 ∈ Pm,r(t) and for 0 ≤ n ≤ 47.
By Lemma7.3 we conclude thatp{3,3}(25n+t0)≡0 (mod 5) for allt0 ∈Pm,r(t) and for all n≥0. This completes the proof of the theorem.
Bibliography
[1] Zakir Ahmed and Nayandeep Deka Baruah. New congruences for Andrews’
singular overpartitions. Int. J. Number Theory, 11(7):2247–2264, 2015.
[2] George E. Andrews. Singular overpartitions. Int. J. Number Theory, 11(5):1523–1533, 2015.
[3] George E. Andrews and F. G. Garvan. Dyson’s crank of a partition. Bull.
Amer. Math. Soc. (N.S.), 18(2):167–171, 1988.
[4] George E. Andrews and David Newman. The minimal excludant in integer partitions. J. Integer Seq., 23(2):Art. 20.2.3, 11, 2020.
[5] Victor Manuel Aricheta. Congruences for Andrews’ (k, i)-singular overparti- tions. Ramanujan J., 43(3):535–549, 2017.
[6] A. O. L. Atkin. Proof of a conjecture of Ramanujan. Glasgow Math. J., 8:14–32, 1967.
[7] Rupam Barman and Chiranjit Ray. Congruences for `-regular overpartitions and Andrews’ singular overpartitions. Ramanujan J., 45(2):497–515, 2018.
[8] Rupam Barman and Chiranjit Ray. Divisibility of Andrews’ singular overpar-
114 Bibliography
[9] Bruce C. Berndt. Ramanujan’s notebooks. Part III. Springer-Verlag, New York, 1991.
[10] Bruce C. Berndt. Number theory in the spirit of Ramanujan, volume 34 of Student Mathematical Library. American Mathematical Society, Providence, RI, 2006.
[11] Anthony J. F. Biagioli. The construction of modular forms as products of transforms of the Dedekind eta function. Acta Arith., 54(4):273–300, 1990.
[12] Shi-Chao Chen, Michael D. Hirschhorn, and James A. Sellers. Arithmetic prop- erties of Andrews’ singular overpartitions. Int. J. Number Theory, 11(5):1463–
1476, 2015.
[13] S. Chowla. Congruence Properties of Partitions. J. London Math. Soc., 9(4):247, 1934.
[14] Sylvie Corteel and Jeremy Lovejoy. Overpartitions. Trans. Amer. Math. Soc., 356(4):1623–1635, 2004.
[15] Tessa Cotron, Anya Michaelsen, Emily Stamm, and Weitao Zhu. Lacunary eta-quotients modulo powers of primes. Ramanujan J., 53(2):269–284, 2020.
[16] Robson da Silva and James A. Sellers. Parity considerations for the mex-related partition functions of Andrews and Newman. J. Integer Seq., 23(5):Art. 20.5.7, 9, 2020.
[17] Robson da Silva and James A. Sellers. Arithmetic properties of 3-regular par- titions in three colours. Bull. Aust. Math. Soc., 104(3):415–423, 2021.
[18] Fred Diamond and Jerry Shurman. A first course in modular forms, volume 228 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2005.
[19] F. J. Dyson. Some guesses in the theory of partitions. Eureka, (8):10–15, 1944.
Bibliography 115
[20] Leonhard Euler. Introduction to analysis of the infinite. Book I. Springer- Verlag, New York, 1988. Translated from the Latin and with an introduction by John D. Blanton.
[21] Aviezri S. Fraenkel and Udi Peled. Harnessing the unwieldy MEX function. In Games of no chance 4, volume 63 of Math. Sci. Res. Inst. Publ., pages 77–94.
Cambridge Univ. Press, New York, 2015.
[22] D. S. Gireesh and M. S. Mahadeva Naika. On 3-regular partitions in 3-colors.
Indian J. Pure Appl. Math., 50(1):137–148, 2019.
[23] Basil Gordon and Kim Hughes. Multiplicative properties of η-products. II. In A tribute to Emil Grosswald: number theory and related analysis, volume 143 of Contemp. Math., pages 415–430. Amer. Math. Soc., Providence, RI, 1993.
[24] Basil Gordon and Ken Ono. Divisibility of certain partition functions by powers of primes. Ramanujan J., 1(1):25–34, 1997.
[25] G. H. Hardy and S. Ramanujan. Asymptotic formulæin combinatory analysis [Proc. London Math. Soc. (2) 17(1918), 75–115]. In Collected papers of Srini- vasa Ramanujan, pages 276–309. AMS Chelsea Publ., Providence, RI, 2000.
[26] Michael D. Hirschhorn. Partitions in 3 colours. Ramanujan J., 45(2):399–411, 2018.
[27] Samuel D. Judge, William J. Keith, and Fabrizio Zanello. On the density of the odd values of the partition function. Ann. Comb., 22(3):583–600, 2018.
[28] T. Kathiravan. Ramanujan-type congruences modulo m for (l, m)-regular bi- partitions. Indian J Pure Appl Math, 53:375–391, 2022.
[29] William J. Keith and Fabrizio Zanello. Parity of the coefficients of certain
116 Bibliography
[30] Byungchan Kim. The overpartition function modulo 128. Integers, 8:A38, 8, 2008.
[31] Neal Koblitz. Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.
[32] Edmund Landau. Uber die einteilung der positiven ganzen zahlen in vier klassen nach der mindestzahl der zu ihrer additiven zusammensetzun erforderlichen quadrate. Arch. Math. Phys., 13:305–312, 1908.
[33] Gottfried Wilhelm Leibniz. S¨amtliche Schriften und Briefe. Reihe VII.
Akademie-Verlag, Berlin, 1990. Mathematische Schriften. Erster Band. 1672–
1676. [Mathematical writings. Vol. 1. 1672–1676], Geometrie—Zahlentheorie—
Algebra (I. Teil). [Geometry—number theory—algebra (Part I)], With a fore- word by Albert Heinekamp, Edited by E. Knobloch and Walter S. Contro.
[34] Xiaorong Li and Olivia X. M. Yao. New infinite families of congruences for Andrews’ (k, i)-singular overpartitions. Quaest. Math., 41(7):1005–1019, 2018.
[35] G. Ligozat. Courbes modulaires de genre1. Publication Math´ematique d’Orsay, No. 75 7411. U.E.R. Math´ematique, Universit´e Paris XI, Orsay, 1974.
[36] M. S. Mahadeva Naika and D. S. Gireesh. Congruences for Andrews’ singular overpartitions. J. Number Theory, 165:109–130, 2016.
[37] Karl Mahlburg. The overpartition function modulo small powers of 2. Discrete Math., 286(3):263–267, 2004.
[38] Yves Martin. Multiplicative η-quotients. Trans. Amer. Math. Soc., 348(12):4825–4856, 1996.
[39] Morris Newman. Construction and application of a class of modular functions.
Proc. London Math. Soc. (3), 7:334–350, 1957.
Bibliography 117
[40] Morris Newman. Construction and application of a class of modular functions.
II. Proc. London Math. Soc. (3), 9:373–387, 1959.
[41] Ken Ono. Distribution of the partition function modulo m. Ann. of Math. (2), 151(1):293–307, 2000.
[42] Ken Ono. The web of modularity: arithmetic of the coefficients of modular forms andq-series, volume 102 ofCBMS Regional Conference Series in Mathematics.
Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.
[43] Ken Ono and Yuichiro Taguchi. 2-adic properties of certain modular forms and their applications to arithmetic functions. Int. J. Number Theory, 1(1):75–101, 2005.
[44] Thomas R. Parkin and Daniel Shanks. On the distribution of parity in the partition function. Math. Comp., 21:466–480, 1967.
[45] Utpal Pore and S. N. Fathima. Some congruences modulo 2, 8 and 12 for Andrews’ singular overpartitions. Note Mat., 38(1):101–114, 2018.
[46] Hans Rademacher. On the Partition Function p(n). Proc. London Math. Soc.
(2), 43(4):241–254, 1937.
[47] Silviu Radu. An algorithmic approach to Ramanujan’s congruences. Ramanujan J., 20(2):215–251, 2009.
[48] Silviu Radu and James A. Sellers. Congruence properties modulo 5 and 7 for the pod function. Int. J. Number Theory, 7(8):2249–2259, 2011.
[49] Silviu Radu and James A. Sellers. Parity results for broken k-diamond parti- tions and (2k+ 1)-cores. Acta Arith., 146(1):43–52, 2011.
118 Bibliography
[50] S. Ramanujan. Congruence properties of partitions [Proc. London Math. Soc.
(2) 18 (1920), Records for 13 March 1919]. In Collected papers of Srinivasa Ramanujan, page 230. AMS Chelsea Publ., Providence, RI, 2000.
[51] S. Ramanujan. Some properties of p(n), the number of partitions of n [Proc.
Cambridge Philos. Soc. 19 (1919), 207–210]. In Collected papers of Srinivasa Ramanujan, pages 210–213. AMS Chelsea Publ., Providence, RI, 2000.
[52] Atle Selberg. Reflections around the Ramanujan centenary. Normat, 37(1):2–7, 43, 1989.
[53] Jean-Pierre Serre. Divisibilit´e des coefficients des formes modulaires de poids entier. C. R. Acad. Sci. Paris S´er. A, 279:679–682, 1974.
[54] Jean-Pierre Serre. Divisibilit´e de certaines fonctions arithm´etiques. In S´eminaire Delange-Pisot-Poitou, 16e ann´ee (1974/75), Th´eorie des nombres, Fasc. 1, Exp. No. 20, page 28. 1975.
[55] Jean-Pierre Serre. Valeurs propres des op´erateurs de Hecke modulo l. In Journ´ees Arithm´etiques de Bordeaux (Conf., Univ. Bordeaux, 1974), pages 109–
117. Ast´erisque, Nos. 24–25. 1975.
[56] Jean-Pierre Serre. Divisibilit´e de certaines fonctions arithm´etiques. Enseign.
Math. (2), 22(3-4):227–260, 1976.
[57] Jacob Sturm. On the congruence of modular forms. In Number theory (New York, 1984–1985), volume 1240 of Lecture Notes in Math., pages 275–280.
Springer, Berlin, 1987.
[58] John Tate. The non-existence of certain Galois extensions of Q unramified outside 2. InArithmetic geometry (Tempe, AZ, 1993), volume 174 ofContemp.
Math., pages 153–156. Amer. Math. Soc., Providence, RI, 1994.
Bibliography 119
[59] Stephanie Treneer. Congruences for the coefficients of weakly holomorphic modular forms. Proc. London Math. Soc. (3), 93(2):304–324, 2006.
[60] Liuquan Wang. Arithmetic properties of (k, `)-regular bipartitions. Bull. Aust.
Math. Soc., 95(3):353–364, 2017.
[61] G. N. Watson. Ramanujans Vermutung ¨uber Zerf¨allungszahlen. J. Reine Angew. Math., 179:97–128, 1938.
[62] Ernest X. W. Xia and X. M. Yao. Some modular relations for the G¨ollnitz- Gordon functions by an even-odd method. J. Math. Anal. Appl., 387(1):126–
138, 2012.
[63] Xinhua Xiong. Overpartitions and ternary quadratic forms. Ramanujan J., 42(2):429–442, 2017.
[64] Fanggang Xue and Olivia X. M. Yao. Explicit congruences modulo 2048 for overpartitions. Ramanujan J., 54(1):63–77, 2021.
120 Bibliography