For $\kappa >1$ and $t,X\geq 1$ $$\sum _{n\leq X}a_n=\frac {1}{2\pi i}\int _{\kappa \pm iT}\frac {\mathcal F(s)X^sds}{s}+\mathcal O\left (x^\kappa \sum _{=1}^\infty \frac {1}{n^\kappa (1+T|\log (X/n)|)}\right )$$ where $a_n\ll 1$ and $$\mathcal F(s)=\sum _{n=1}^\infty \frac {a_n}{n^s}.$$ This is a quantitative version of Perron's formula - the above is taken from Page 132 of Tenenbaum's *Introduction to Probabilistic Number Theory*.

On page 134, display (11), we have a qualitative version of Perron wit the first Cesaro weight $$\sum _{n\leq X}a_n(X-n)=\frac {1}{2\pi i}\int _{\kappa \pm i\infty }\frac {\mathcal F(s)X^{s+1}ds}{s(s+1)}.$$

Is there anywhere a quantative version of this weighted version? I can prove it, but surely it must already be somewhere.