On the number of even values of an eta-quotient

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Department of Mathematical Sciences


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 ga,b(x) denotes the number of even coefficients of F in degrees n b (mod a) such that n ≤ x, then we show that ga,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, xloglog 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.

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International Journal of Number Theory