Logarithmic coefficient bound in Vaughan identity ($|{\sum }_{cd=m,c\le U,d\le V}\mu (c)\Lambda (d)|\le \log m$)

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 $\log N$.