The Ramanujan-Dyson identities and George Beck's congruence conjectures

Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson’s rank. George Beck has generalized the study of partition function congruences related to rank by considering the total number of parts in the partitions of n. The related generating functions are no longer part of the world of mock theta functions. However, George Beck has conjectured that certain linear combinations of the related enumeration functions do satisfy congruences modulo 5 and 7. The conjectures are proved here.

"Electronic version of an article published as International Journal of Number Theory Vol. 17, No 02, oo. 239-249 (2021), 10.1142/S1793042120400060 © World Scientific Publishing Company https://doi.org/10.1142/S1793042120400060

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Work Title The Ramanujan-Dyson identities and George Beck's congruence conjectures
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Open Access
Creators
  1. George E Andrews
Keyword
  1. conjectures
  2. partial function
  3. Ramanujan congruences
License In Copyright (Rights Reserved)
Work Type Article
Publication Date June 23, 2020
Publisher Identifier (DOI)
  1. 10.1142/S1793042120400060
Deposited July 19, 2021

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    Keyword
    • conjectures, partial function, Ramanujan congruences
    Publisher Identifier (DOI)
    • 10.1142/S1793042120400060
    Description
    • Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson’s rank. George Beck has generalized the study of partition function congruences related to rank by considering the total number of parts in the partitions of n. The related generating functions are no longer part of the world of mock theta functions. However, George Beck has conjectured that certain linear combinations of the related enumeration functions do satisfy congruences modulo 5 and 7. The conjectures are proved here.
    Publisher's Statement
    • "Electronic version of an article published as International Journal of Number Theory Vol. 17, No 02, oo. 239-249 (2021), 10.1142/S1793042120400060 © World Scientific Publishing Company
    • https://doi.org/10.1142/S1793042120400060
  • Added Creator George E Andrews
  • Added The Ramanujan-Dyson Identities.pdf
  • Updated License Show Changes
    License
    • https://rightsstatements.org/page/InC/1.0/
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