On the number of even values of an eta-quotient

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