Lacunary eta-quotients modulo powers of primes
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Lacunary eta-quotients modulo powers of primes Tessa Cotron1 · Anya Michaelsen2
· Emily Stamm3 · Weitao Zhu2
Received: 23 August 2018 / Accepted: 28 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract An integral power series is called lacunary modulo M if almost all of its coefficients are divisible by M. Motivated by the parity problem for the partition function, Gordon and Ono studied the generating functions for t-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Okounkov, and Han. Keywords Partitions · Eta-quotients · Nekrasov Okounkov formula · Lacunary · Eta-function · Modular forms · Generalized eta-quotients · Powers of primes Mathematics Subject Classification 11P02
The authors were supported by Emory University, the Templeton World Charity Foundation, and the NSF via Grant DMS-1557690.
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Anya Michaelsen [email protected] Tessa Cotron [email protected] Emily Stamm [email protected] Weitao Zhu [email protected]
1
Department of Mathematics, Emory University, 201 Dowman Drive, Atlanta, GA 30322, USA
2
Department of Mathematics, Williams College, 33 Stetson, Williamstown, MA 01267, USA
3
Department of Mathematics, Vassar College, 124 Raymond Ave, Poughkeepsie, NY 12604, USA
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T. Cotron et al.
1 Introduction A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. For example, the set of partitions of 4 is {4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1}. The partition function p(n) counts the number of partitions of n. From the above example we see that p(4) = 5. The generating function for p(n) satisfies the identity P(q) :=
∞ n=0
p(n)q n =
∞ n=1
1 . (1 − q n )
The partition function has many congruence properties modulo primes and powers of primes, the most famous of which are the Ramanujan congruences [11]: p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), p(11n + 6) ≡ 0 (mod 11) for all n ≥ 0. Few results of this form were known until Ahlgren and Ono [1,8] showed that there are infinitely many congruences of the form p(An + B) ≡ 0 (mod m) for any integer m relatively prime to 6. Although no analogous theorem exists for the partition function modulo the primes 2 and 3, Parkin and Shanks conjectured that half of the values for p(n) are even and half are odd [10]. To be precise, given an integral power series F(q) := n−∞ a(n)q n , we define δ(F, M; X ) :=
#{n ≤ X : a(n) ≡ 0 (mod M)} . X
It is conjectured that δ(P, 2; X ) and δ(P, 3; X ) tend to 21 and 13 , respectively, as X approaches infinity. Table 1 contains values of δ(P, 2; X ) and δ(P, 3; X ) for X up to 500,000. Although calculations strongly support this conjecture, it remains unproven. Moreover, it remains unknown whether a pos
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