论文标题
关于eTa Quortient的偶数值的数量
On the number of even values of an eta-quotient
论文作者
论文摘要
本注释的目的是在任何算术进展中都对任意ETA质量$ f $的傅立叶系数的偶数值数量提供一般的下限。也就是说，如果$ g_ {a，b}（x）$表示$ f $ in Be n g $ n \ equiv b $（mod $ a $）的均匀系数的数量，则我们表明$ n \ le x $，那么我们表明$ g_ {a，b}（x） / \ sqrt {x} $ $ x $ x $ x $ x $ x $ x $ x $ x $。 请注意，即使在分区函数的特殊情况下，我们的结果也非常接近当前已知的最佳界限$ p（n）$（即$ \ sqrt {x} \ log \ log \ log x $，在2016年由Bellaïche和Nicolas证明）。我们的论点基本上依赖并概括了Serre对$ p（n）$的偶数值的经典定理，并在某些Eta- Quotients的lacunarity Modulo 2上由Cotron \ emph {et al。）结合了最新的模块化形式。 有趣的是，即使在Serre首先显示的$ P（n）$的情况下，也没有任何基本证明。最后，我们在二次表示上提出了一个优雅的问题，其解决方案最终将产生Serre定理的无模块化证明。
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 $n\equiv b$ (mod $a$) such that $n\le x$, then we show that $g_{a,b}(x) / \sqrt{x}$ 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, $\sqrt{x}\log \log x$, 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 \emph{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.
