= Logarithmically smoothed Perron formula
{title2=$\sum_{n\le x}a_n\log(x/n)$}
If the <Dirichlet series> $F(s)=\sum_na_nn^{-s}$ is absolutely convergent at $\sigma>0$, then
$$
\sum_{n\le x}a_n\log(x/n)
=\frac1{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{F(s)x^s}{s^2}\,ds
=\frac1{2\pi}\int_{\mathbb R}\frac{F(\sigma+it)x^{\sigma+it}}{(\sigma+it)^2}\,dt.
$$
The scalar kernel is $(2\pi)^{-1}\int_{\mathbb R}e^{v(\sigma+it)}(\sigma+it)^{-2}dt=v_+$. This follows by <Fourier inversion> applied to $u e^{-\sigma u}\mathbf1_{u\ge0}$, whose <Fourier transform> is $(\sigma+it)^{-2}$. Absolute convergence of the series and of $\int |\sigma+it|^{-2}dt$ justifies interchange. The factor $ds=i\,dt$ is essential.
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