Abstract
We find and prove a class of congruences modulo 4 for Andrews’ partition with certain ternary quadratic form. We also discuss distribution of \(\overline{\mathcal{E}\mathcal{O}}(n)\) and further prove that \(\overline{\mathcal{E}\mathcal{O}}(n)\equiv 0\pmod 4\) for almost all n. This study was inspired by similar congruences modulo 4 in the work by the second author and Garvan.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions. Ann. Comb. 22 (2018), no. 3, 433–445.
B. C. Berndt, Ramanujan’s Notebooks, Part III (Springer, New York, NY, 1991).
A. Berkovich and F. Patane, Essentially unique representations by certain ternary quadratic forms. Exp. Math. 24 (2015), no. 1, 8–22.
R. Chen and F. Garvan, Congruences modulo 4 for weight \(3/2\) eta-products. Bull. Aust. Math. Soc. 103 (2021), no. 3, 405–417.
R. Chen and F. Garvan, A proof of the mod 4 unimodal sequence conjectures and related mock theta functions, Adv. Math. 398 (2022), Paper No. 108235, 50 pp.
S. Cooper and H. Y. Lam, On the Diophantine equation \(n^2 = x^2 + by^2 + cz^2\), J. Number Theory 133 (2013), 719–737.
D. A. Cox, Primes of the form \(x^2 + ny^2\), Fermat, class field theory and complex multiplication. Wiley, New York, 1989.
X. J. Guo, Y. Z. Peng and H. R. Qin, On the representation numbers of ternary quadratic forms and modular forms of weight \(3/2\). J. Number Theory 140 (2014), 235–266.
A. Pizer, On the \(2\) -part of the class number of imaginary quadratic number fields. J. Number Theory 8 (1976), no. 2, 184–192.
C. Ray and R. Barman, On Andrews’ integer partitions with even parts below odd parts. J. Number Theory 215 (2020), 321–338.
T. R. Shemanske, Representations of ternary quadratic forms and the class number of imaginary quadratic fields, Pacific J. Math. 122(1) (1986), 223–250.
L. Wang, Parity of coefficients of mock theta functions. J. Number Theory 229 (2021), 53–99.
Acknowledgements
The first author was supported in part by Shanghai Sailing Program (#21YF1413600) and the National Natural Science Foundation of China (Grant No. 12201387). The second author was supported in part by the Postdoctoral Science Foundation of China (#2022M712422).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Frédérique Bassino.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chen, D., Chen, R. Congruence Modulo 4 for Andrews’ Even Parts Below Odd Parts Partition Function. Ann. Comb. 27, 269–279 (2023). https://doi.org/10.1007/s00026-023-00645-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-023-00645-3