1. Parity of the coefficients of certain eta-quotients, II: The case of even-regular partitions.
- Author
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Keith, William J. and Zanello, Fabrizio
- Subjects
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PARTITION functions , *MODULAR forms - Abstract
We continue our study of the density of the odd values of eta-quotients, here focusing on the m -regular partition functions b m for m even. Based on extensive computational evidence, we propose an elegant conjecture which, in particular, completely classifies such densities: Let m = 2 j m 0 with m 0 odd. If 2 j < m 0 , then the odd density of b m is 1/2; moreover, such density is equal to 1/2 on every (nonconstant) subprogression A n + B. If 2 j > m 0 , then b m , which is already known to have density zero, is identically even on infinitely many non-nested subprogressions. This and all other conjectures of this paper are consistent with our "master conjecture" on eta-quotients presented in the previous work. In general, our results on b m for m even determine behaviors considerably different from the case of m odd. Also interesting, it frequently happens that on subprogressions A n + B , b m matches the parity of the multipartition functions p t , for certain values of t. We make a suitable use of Ramanujan-Kolberg identities to deduce a large class of such results; as an example, b 28 (49 n + 12) ≡ p 3 (7 n + 2) (mod 2). Additional consequences are several "almost always congruences" for various b m , as well as new parity results specifically for b 11. We wrap up our work with a much simpler proof of the main result of a recent paper by Cherubini-Mercuri, which fully characterized the parity of b 8. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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