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Large finite products of small fractions (1908.00839v2)
Published 2 Aug 2019 in math.CA
Abstract: Fix positive reals $a,b,c,d$, and let $h(x)$ be a real function behaving sort of like $\sin x$ near 0. Then, provided $m$ grows linearly with $n$. there exists a positive constant $C$ such that$$ \prod_{j=0}m\frac{h\left((cj+a)\frac{d}{n}\right)}{h\left((cj+b)\frac{d}{n}\right)}\sim C n{\frac{a-b}c}. $$
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