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nt.quantity concept – Quantitative Perron components with weights

For $kappa >1$ and $t,Xgeq 1$ $$sum _{nleq X}a_n=frac {1}{2pi i}int _{kappa pm iT}frac {mathcal F(s)X^sds}{s}+mathcal Oleft (x^kappa sum _{=1}^infty frac {1}log (X/n)privilege )$$ the place $a_nll 1$ and $$mathcal F(s)=sum _{n=1}^infty frac {a_n}{n^s}.$$ This is a quantitative model of Perron’s components – the above is taken from Page 132 of Tenenbaum’s *Introduction to Probabilistic Number Theory*.

On web page 134, array (11), now we have a qualitative model of Perron humor the primary Cesaro weight $$sum _{nleq X}a_n(X-n)=frac {1}{2pi i}int _{kappa pm iinfty }frac {mathcal F(s)X^{s+1}ds}{s(s+1)}.$$

Is there wherever a quantative model of this weighted model? I can show it, however absolutely it should already breathe someplace.

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