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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]

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of m -regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all m ≤ 28 , we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area. [ABSTRACT FROM AUTHOR]

A famous conjecture of Parkin-Shanks predicts that p (n) is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the "striking" fact that, if for any A ≡ ± 1 (mod 6) the multipartition function p A (n) has positive odd density, then so does p (n). Similarly, the positive odd density of any p A (n) with A ≡ 3 (mod 6) would imply that of p 3 (n). Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture. [ABSTRACT FROM AUTHOR]

We continue our study of a basic but seemingly intractable problem in integer partition theory, namely the conjecture that p ( n ) is odd exactly 50% of the time. Here, we greatly extend on our previous paper by providing a doubly-indexed, infinite framework of conjectural identities modulo 2, and show how to, in principle, prove each such identity. However, our conjecture remains open in full generality. A striking consequence is that, under suitable existence conditions, if any t -multipartition function is odd with positive density and t ≢ 0 ( mod 3 ) , then p ( n ) is also odd with positive density. These are all facts that appear virtually impossible to show unconditionally today. Our arguments employ a combination of algebraic and analytic methods, including certain technical tools recently developed by Radu in his study of the parity of the Fourier coefficients of modular forms. [ABSTRACT FROM AUTHOR]

Using entirely algebraic or elementary methods, partly inspired by recent ideas of P. Monsky on quadratic representations, we determine a new asymptotic lower bound for the number of odd values of a fundamental modular function in number theory, the Klein j -function. Namely, the number of n ≤ x such that the Klein j -function — or equivalently, the cubic partition function — is odd has order at least x log ⁡ log ⁡ x log ⁡ x , for x → ∞ . This improves recent results of Berndt, Yee and Zaharescu and Chen and Lin, approaching significantly the best bound currently known for p ( n ) , obtained using modular forms. Then, in the final section, we show how to employ modular forms to slightly refine our bound. In fact, our brief argument, combining a recent result of Nicolas–Serre with a classical theorem of Serre on level 1 modular forms, more generally provides a lower bound for the odd values of any positive power of the generating function of p ( n ) . [ABSTRACT FROM AUTHOR]