Parity of the coefficients of certain eta-quotients, II: The case of even-regular partitions

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Department of Mathematical Sciences


We continue our study of the density of the odd values of eta-quotients, here focusing on the m-regular partition functions bm for m even. Based on extensive computational evidence, we propose an elegant conjecture which, in particular, completely classifies such densities: Let m=2jm0 with m0 odd. If 2jm0, then bm, 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 bm for m even determine behaviors considerably different from the case of m odd. Also interesting, it frequently happens that on subprogressions An+B, bm matches the parity of the multipartition functions pt, 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, b28(49n+12)≡p3(7n+2)(mod2). Additional consequences are several “almost always congruences” for various bm, as well as new parity results specifically for b11. 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 b8.

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Journal of Number Theory