= Logarithmic coefficient bound in Vaughan identity
{title2=$|\sum_{cd=m,c\le U,d\le V}\mu(c)\Lambda(d)|\le\log m$}
The <Von Mangoldt function> is nonnegative and satisfies $\sum_{d\mid m}\Lambda(d)=\log m$. Therefore the middle short-divisor coefficient in the <Vaughan identity> has the displayed pointwise bound. Grouping its long-divisor term as $\mu_{>U}(m)b(n)$, with $b(n)=\sum_{d\mid n,d>V}\Lambda(d)$, gives $0\le b(n)\le\log n$. This grouping permits bounded-coefficient <Type II sums> after division by $\log N$.
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