$\mathcal M\widetilde{1_\varepsilon}(p)=1/p+O(\varepsilon)$ for $1/2\le\operatorname{Re}p\le1$

Throughout, $\nu$ is a fixed smoothing kernel: a $C^1$ function on $\mathbb R$ supported in $[1/2,2]$, nonnegative on $(0,\infty)$, with $\int_0^\infty\nu(x)\,dx/x=1$; $\widetilde{1_\varepsilon}=$ Smooth1 ν ε is the smoothed indicator of $(0,1]$ obtained by Mellin convolution with the delta-spike $\nu(x^{1/\varepsilon})/\varepsilon$ (it equals $1$ on $(0,1-\varepsilon\log2]$, $0$ on $[1+2\varepsilon\log 2,\infty)$, and lies in $[0,1]$), and $\mathcal M\widetilde{1_\varepsilon}(s)=\int_0^\infty\widetilde{1_\varepsilon}(x)x^{s-1}dx$ is its Mellin transform (Mathlib's mellin).

Statement. There is a constant $C>0$ (depending only on $\nu$) such that for all $0<\varepsilon<1$ and all $p\in\mathbb C$ with $1/2\le\operatorname{Re}p\le1$,

Indeed $\mathcal M\widetilde{1_\varepsilon}(p)=p^{-1}\,\mathcal M\nu(\varepsilon p)$ with $\mathcal M\nu(w)=\int_{1/2}^{2}\nu(x)x^{w-1}dx$, and $|x^{\varepsilon p}-1|\ll\varepsilon|p|$ on $[1/2,2]$ when $\varepsilon|p|\le1$, while for $\varepsilon|p|>1$ both $\mathcal M\widetilde{1_\varepsilon}(p)\ll1/(\varepsilon|p|^2)$ and $1/|p|$ are $\ll\varepsilon$. It converts the residues $r(p)\,\mathcal M\widetilde{1_\varepsilon}(p)X^{p}$ produced by the contour method into the terms $r(p)X^{p}/p$ of the final asymptotic formula (the main term $X$ and the exceptional term $-X^{\beta}/\beta$), at the cost $O(\varepsilon X)$.