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Abstract
We investigate arithmetic properties of an overpartition counting function that first
arose in connection with Schur’s partition theorem and a universal mock theta function.
Motivated by work of Basil Gordon on the Rogers–Ramanujan identities, we first give a
complete characterization of the parity of this overpartition function in the progressions
2 n
+ 1 ,
4 n
+ 2 , and
8 n
+ 4 in terms of the
factorization of
A n
+
B
for certain
A
and
B .
We then find similar characterizations of the residue modulo
4 in the
progressions
8 n
+ 5
and
8 n
+ 7 .
Finally, we prove some Ramanujan-type congruences modulo
5 . Our
proofs use basic facts about modular forms and some elementary algebraic number
theory.
Keywords
partitions, overpartitions, parity, eta-quotients, binary
quadratic forms
Mathematical Subject Classification
Primary: 11P83
Secondary: 11E25, 11P84
Milestones
Received: 2 August 2021
Revised: 16 November 2021
Accepted: 18 November 2021
Published: 2 December 2022
Communicated by Ken Ono