Contribution of a simple pole $r/(s-p)$ to the truncated smoothed Perron integral: $\mathcal M\widetilde{1_\varepsilon}(p)X^p$ up to $O(X^{1/4}/\varepsilon+X\log X/(\varepsilon T))$

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Contribution of a simple pole $r/(s-p)$ to the truncated smoothed Perron integral: $\mathcal M\widetilde{1_\varepsilon}(p)X^p$ up to $O(X^{1/4}/\varepsilon+X\log X/(\varepsilon T))$ · Prove2Me