ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3

ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3 JOHN J WEBB Abstract. Let b 13 n) denote the number of 13-regular partitions of n. We study in this paper the behavior of b 13 n) modulo 3 where n 1 mod 3). In particular, we identify an infinite family of non-nested arithmetic progressions modulo arbitray powers of 3 such that b 13 n) 0 mod 3), confirming a conjecture in [5]. 1. Introduction Let pn) be the number of partitions of a positive integer n: the number of non-increasing sequences of positive integers whose sum is n. By convention, we agree that p0) := 1. Partitions play an important role in number theory, combinatorics, representation theory, and mathematical physics. Euler showed that the generating function for pn) satisfies pn)q n = 1 q m ) 1. m=1 Among the most famous results on pn) are the Ramanujan congruences. For all non-negative integers n, these are p5n + 4) 0 mod 5), p7n + 5) 0 mod 7), p11n + 6) 0 mod 11). Recent works of Ahlgren and Ono, [9], [1], and [3], show that for all moduli m coprime to 6, there are infinitely many non-nested arithmetic progressions An + B such that pan + B) 0 mod m). These works motivate an investigation of congruence phenomena for other types of partition functions. Let k be a positive integer. In this paper, we study congruences for k-regular partitions, which we now define. We say that a partition of n is k-regular if none of its parts are divisible by k. We let b k n) denote the number of k-regular partitions of n, and we agree that b k 0) := 1. The generating function for b k n) is 1.1) b k n)q n = m=1 1 q km ) 1 q m ). We now briefly describe some of the known facts on divisibility and distribution of b k n) mod m). Gordon and Ono in [7] prove the following result. Let k be a positive integer and Date: October 8, 009 Final Draft. 1991 Mathematics Subject Classification. 11P83. 1

JOHN J WEBB let p be a prime divisor of k. If p ordp k k, then for every positive integer j, the arithmetic density of integers n for which b k n) 0 mod p j ) is 1. In [1] and [13], Penniston proves results on the distribution of b k n) in residue classes modulo p i. Specifically, in [13], if k p are primes with 3 k 3 and p 5, then the density of positive integers n such that b k n) 0 mod p) is at least p+1 p if p k 1 and at least p 1 p if p k 1. In [8] and [6], the authors use the theory of complex multiplication to precisely describe those n for which b l n) 0 mod l) when l {3, 5, 7, 11}. We also recall the classical result of Euler which states that b n), the number of partitions into odd parts, agrees with the number of partitions of n into distinct parts, often denoted by Qn). Recent works on the arithmetic of Qn) include, for example, [] and [11]. In this paper, we prove a conjecture made in the seven-author paper [5] on b 13 n) mod 3). In [5], the authors prove that there are infinitely many Ramanujan-type congruences modulo for b 5 n) and b 13 n) in arithmetic progressions of even numbers. Additionally, they prove that for every n 0, b 5 0n + 5) 0 mod ) and b 5 0n + 13) 0 mod ). Further, the authors in [5] make the following conjecture about b 13 n). Conjecture 1 [5]). For all integers n and l with n 0 and l, we have b 13 3 l n + 5 ) 3l 1 1 0 mod 3). The authors verified their conjecture for l 6. We prove the following theorem on b 13 n) modulo 3 for n in certain arithmetic progressions modulo powers of 3. Theorem 1.1. For integers n and l with n 0 and l 1, we have ) b 13 3n + 1) 1) l+1 b 13 3 l n + 3l 1 mod 3). Then, using Theorem 1.1 we prove Theorem 1.. Conjecture 1 is true. To prove Theorems 1.1 and 1., we realize the generating function for b 13 3n + 1) as a modular form and study its image under certain operators. In Section, we give the necessary background on modular forms. In Section 3, we prove the theorems.. Background on Modular Forms Let h be the complex upper-half plane, and let q := e πiz. For integers k 0 and N 1, and for χ, a Dirichlet character modulo N, we let M k Γ 0 N), χ) denote the C-vector space of weight k holomorphic modular forms on Γ 0 N) with character χ. We require Dedekind s eta-function, defined by := q 1/4 n=1 1 q n ).

ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3 3 It serves as an important building block for modular forms, as we now describe. Let N be a positive integer. We say that a function fz) is an eta-quotient if it has the form fz) = ηδz) r δ, where each r δ is an integer. One may apply the following theorem of Gordon, Hughes, and Newman to determine whether an eta-quotient transforms properly with respect to the congruence subgroup Γ 0 N). Theorem.1 Gordon, Hughes, and Newman, [10]). If fz) = with k = 1 r δ Z with the additional properties that and δr δ 0 mod 4) N r δ δ 0 mod 4), ηδz) r δ is an eta-quotient then fz) satisfies ) a b for every c d s := δ r δ. f ) az + b = χd)cz + d) k fz) cz + d Γ 0 N). Here the character χ is defined by χd) := ) 1) k s, where d We note that is holomorphic and non-zero on h. Therefore, the poles of an eta-quotient satisfying Theorem.1, if it has any, must be supported at cusps. One may use the following theorem of Ligozat to compute the orders of vanishing at cusps for eta-quotients satisfying Theorem.1. If all such orders are non-negative then the eta-quotient is a holomorphic modular form. Theorem. Ligozat, [10]). Let c, d, and N be positive integers with d N and gcdc, d) = 1. If fz) is an eta-quotient satisfying the conditions of Theorem.1 for N, then the order of vanishing of fz) at the cusp c is d N gcdd, δ) r δ 4 gcdd, N )dδ. d For example, Theorems.1 and. imply that.1) η13z) 37 M 18 Γ 0 13), χ 13 ), where χ 13 is the Kronecker character associated to Q 13). We require this modular form for the proofs of Theorems 1.1 and 1.. Next, we require certain standard operators on

4 JOHN J WEBB modular forms. For positive integers d, we define the operator U d on formal power series in q by an)q n U d := adn)q n. When acting on spaces of modular forms, if d N, we have U d : M k Γ 0 N), χ) M k Γ 0 N), χ). The U d operator has the additional factorization property that [ ) )] ) ).) an)q dn bm)q m U d = an)q n bdm)q m. m=0 For positive integers m, the Hecke operator T m is an endomorphism on M k Γ 0 N), χ). Let p be prime. Then for fz) = an)qn M k Γ 0 N), χ) we have f T p := apn) + χp)p k 1 an/p) ) q n, where if p n we agree that an/p) = 0. In the setting of modular forms with p-integral coefficients of weight greater than 1, it follows that T p and U p agree modulo p. We also recall the notion of twisting. Let ε be a Dirichlet character modulo M, and let fz) = an)qn. Then we define the twist of f by ε as fz) ε := εn)an)q n. If fz) M k Γ 0 N), χ), let N be the conductor of χ, and define N := lcmn, N M, M ). Then we have fz) ε M k Γ 0 N ), ε χ) see for example [4]). We now state an important theorem of Sturm which provides a method to test whether two modular forms with integer coefficients are congruent modulo a prime. Let m be a positive integer, and let fz) = an) be a modular form with integer coefficients. Then we define ord m fz)) to be the smallest n such that an) 0 mod m); if no such n exists, then we say that ord m fz)) :=. Theorem.3 Sturm, [10]). Suppose that N is a positive integer, p is prime and fz), gz) M k Γ 0 N), χ) Z[[q]]. If then we have fz) gz) mod p). ord p fz) gz)) > k 1 [SL Z) : Γ 0 N)], m=0 We note that [SL Z) : Γ 0 N)] = N l prime: l N 1 + 1 ). l

ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3 5 3. Proof of Theorems 1.1 and 1. We begin by realizing the generating function for b 13 3n + 1) as a modular form. For all m Z, define G m z) by 3.1) G m z) := η6z) m 1 η78z) 5 m. Using Theorems.1 and., we see for 0 m 13 that G m z) M 18 Γ 0 31), χ 13 ), where as before χ 13 is the Kronecker character attached to Q 13). We define 3.) Hz) :=G 0 z) + G z) + G 3 z) + G 4 z) + G 5 z) + G 6 z) + G 7 z) + G 8 z) + G 9 z) + G 10 z) + G 11 z) + G 13 z) M 18 Γ 0 31), χ 13 ). Proposition 3.1. We have Hz) b 13 3n + 1)q 6n+3 mod 3). Proof of Proposition 3.1. Using.1), we start with 3.3) η13z) 37 = q 0 1 q 13n ) 36 b 13 n)q n M 18 Γ 0 13), χ 13 ). n 1 The first step is to explicitly identify a form hz) M 18 Γ 0 13), χ 13 ) such that 3.4) hz) η13z)37 For all m Z, define g m z) by T 3 mod 3). 3.5) g m z) := m 1 η13z) 37 m. Then by Theorems.1 and., we have g m z) M 18 Γ 0 13), χ 13 ) for 1 m 0. We define hz) as 3.6) hz) :=g 0 z) + g z) + g 3 z) + g 4 z) + g 5 z) + g 6 z) + g 7 z) + g 8 z) + g 9 z) + g 10 z) + g 11 z) + g 13 z) M 18 Γ 0 13), χ 13 ). Using Sturm s Theorem, we see that this hz) satisfies 3.4). On the other hand, using.), 3.3), and the fact that the T 3 and U 3 operators agree modulo 3 on modular forms with integer coefficients and weights greater than 1, we find that η13z) 37 T 3 [ ] 1 q 13n ) 1 b 13 n)q n+0 T 3 n 1 3.7) 1 q 13n ) 1 b 13 3n + 1)q n+7 mod 3). n 1 Hence, combining 3.4) and 3.7), we see that 3.8) hz) q 7 n 1 1 q13n ) b 1 13 3n + 1)q n mod 3).

6 JOHN J WEBB We also note from 3.1) and 3.5) that 3.9) G m z) = g m 6z) q 39 n 1 1 q78n ) 1. Thus, combining 3.), 3.6), 3.8) and 3.9), we see that Hz) b 13 3n + 1)q 6n+3 mod 3). The next proposition gives the image modulo 3 of H T 3 under T 3 and twists by characters modulo 3. Proposition 3.. We have 3.10) Hz) T 3 Hz) T 3 mod 3), and 3.11) Hz) T 3 ) χ triv Hz) T 3 ) χ 3 mod 3), where χ triv and χ 3 are respectively the trivial and nontrival characters modulo 3. Proof of Proposition 3.. The validity of the statements follows from calculating enough coefficients of each form and then applying Sturm s Theorem. The forms in 3.10) lie in M 1 Γ 0 31), χ 13 ) whose Sturm bound is 67. Therefore, it suffices to verify that ord 3 Hz) T 3 + Hz) T3 ) > 67. For 3.11), we see that Hz) T 3 ) χ triv and Hz) T 3 ) χ 3 lie in M 1 Γ 0 936), χ 13 ), whose Sturm bound is 016. Thus we verify that ord 3 Hz) T 3 ) χ triv Hz) T 3 ) χ 3 ) > 016. By combining Propositions 3.1 and 3., we now prove Theorems 1.1 and 1.. Proof of Theorems 1.1 and 1.. An immediate consequence of Proposition 3.1 is that 3.1) Hz) T 3 b 13 3n + 1)q n+1 mod 3). Theorem 1.1 follows immediately from iteratively applying 3.10) of Proposition 3. to 3.1) For Theorem 1., 3.11) of Proposition 3. asserts that the nth coefficient of H T 3 is congruent to zero modulo 3 whenever n mod 3). Hence, using 3.1), for all nonnegative integers n we have 3.13) b 13 33n + ) + 1) = b 13 9n + 7) 0 mod 3). By Theorem 1.1 we have ) b 13 9n + 7) = b 13 33n + ) + 1) 1) l+1 b 13 3 l 3n + ) + 3l 1 1) l+1 b 13 3 l+1 + 5 ) 3l 1 3.14). Finally, we combine 3.13) and 3.14) to get our result. mod 3)

ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3 7 Remark. In [5], the authors proved 3.13) by showing that η13z) 37 T 3 0 mod 3). References [1] S. Ahlgren, Distribution of the partition function modulo composite integers M. Math. Ann. 318 000), no. 4, 795 803. [] S. Ahlgren and J. Lovejoy, The Arithmetic of partitions into distinct parts. Mathematika 48 001), no. 1-, 03 11. [3] S. Ahlgren and K. Ono, Congruence properties for the partition function. Proc. Natl. Acad. Sci. USA 98 001), no. 3, 188 1884. [4] A.O.L Atkin and W.C. Li, Twists of newforms and pseudo-eigenvalues of W -operators. Invent. Math. 48 1978), no. 3, 1 43. [5] N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston, J. Radder, Divisibility properties of the 5-regular and 13-regular partition functions. Integers 8 008), no., A60, 10pp. [6] B. Dandurand and D. Penniston, l-divisiblity of l-regular partition functions. Ramanujan J. 19 009), no. 1, 63 70. [7] B. Gordon and K. Ono, Divisibility of certain partition functions by powers of primes. Ramanujan J. 1 1997), no. 1, 5 34. [8] J. Lovejoy and D. Penniston, 3-regular partitions and a modular K3 surface. q-series with applications to combinatorics, number theory, and physics Urbana, IL, 000), 177 18, Contemp. Math., 91, Amer. Math. Soc., Providence, RI, 001. [9] K. Ono, Distribution of the partition function modulo m. Ann. of Math. ) 151 000), no. 1, 93 307. [10] K. Ono, The Web of Modularity. CBMS Regional Conference Series in Mathematics, 10. American Mathematical Society, Providence, RI, 004. [11] K. Ono and D. Penniston, The -adic behavior of the number of partitions into distinct parts. J. Combin. Theory Ser. A 9 000), no., 138 157. [1] D. Penniston, The p a -regular partition function modulo p j. J. Number Theory 94 00), no., 30 35. [13] D. Penniston, Arithmetic of l-regular partition functions. Int. J. Number Theory 4 008), no., 95 30. Department of Mathematics, University of South Carolina, Columbia, SC 908 E-mail address: webbjj3@mailbox.sc.edu