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The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient F , over any arithmetic progression. Namely, if g a , b (x) denotes the number of even coefficients of F in degrees n ≡ b (mod a) such that n ≤ x , then we show that g a , b (x) / x is unbounded for x large. Note that our result is very close to the best bound currently known even in the special case of the partition function p (n) (namely, x log log x , proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre's classical theorem on the number of even values of p (n) , combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients. Interestingly, even in the case of p (n) first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre's theorem. [ABSTRACT FROM AUTHOR]

Recently, Hirschhorn and the first author considered the parity of the function a (n) which counts the number of integer partitions of n wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of a (2 m) based solely on properties of m. In this paper, we quickly reprove their result, and then extend it to an explicit characterization of the parity of a (n) for all n ≢ 7 (mod 8). We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case n ≡ 7 (mod 8) , where, interestingly, the behavior of a (n) modulo 2 appears to be entirely different. In particular, we conjecture that, asymptotically, a (8 m + 7) is odd precisely 5 0 % of the time. This conjecture, whose broad generalization to the context of eta-quotients will be the topic of a subsequent paper, remains wide open. [ABSTRACT FROM AUTHOR]