For a fixed prime $p$, the maximum coefficient (in absolute value) $M(p)$ ofthe cyclotomic polynomial $\Phi_{pqr}(x)$, where $r$ and $q$ are free primessatisfying $r>q>p$ exists. Sister Beiter conjectured in 1968 that$M(p)\le(p+1)/2$. In 2009 Gallot and Moree showed that $M(p)\ge2p(1-\epsilon)/3$ for every $p$ sufficiently large. In this article Kloostermansums (`cloister man sums') and other tools from the distribution of modularinverses are applied to quantify the abundancy of counter-examples to SisterBeiter's conjecture and sharpen the above lower bound for $M(p)$.
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机译：对于固定素数$ p $，环多项式$ \ Phi_ {pqr}（x）$的最大系数（绝对值）$ M（p）$，其中$ r $和$ q $是免费素数满足$ r> q> p $存在。贝特姐妹在1968年推测$ M（p）\ le（p + 1）/ 2 $。 2009年，Gallot和Moree指出，每$ p $足够大，则$ M（p）\ ge2p（1- \ epsilon）/ 3 $。在本文中，Kloostermansums（“ cloister man sum”）和其他来自模块化逆分布的工具被用于量化SisterBeiter猜想的反例的数量，并提高$ M（p）$的下界。
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