= Truncated Perron kernel estimate
{title2=$K_T(v)=(2\pi i)^{-1}\int_{c-iT}^{c+iT}v^s\,ds/s$}
For $v\ne1$, $K_T(v)$ differs from $1_{v>1}$ by $O(v^c\min(1,(T|\log v|)^{-1}))$. Close the contour to the left for $v>1$ and to the right for $v<1$; horizontal sides give the logarithmic denominator and the nearby transition is bounded. At $v=1$, direct integration gives $K_T(1)=\pi^{-1}\arctan(T/c)=1/2+O(c/T)$. This explains the half-weight endpoint in the <truncated Perron formula>. The constants may be taken uniform for $1\le c\le2$ and $T\ge2$.
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