Certain Complex Integral I was trying to generalize the Riemann's prime number formula for $\pi(x)$ to a general algebraic field $K$, and came across the integral:
$$f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{d}{ds}\left[\frac{\log s}{s}\right]x^s ds (x>1, c>0).$$
I have proved through simple calculation that this integral converges and for $\forall \epsilon>0$, $|f(x)|=O(x^\epsilon)$. Can anybody help me, please?
 A: We can express your integral as an inverse Mellin transform or inverse Laplace Transform:
$$ f(x) = \mathcal{M}^{-1}\left[\frac{1}{s^2}-\frac{\log (s)}{s^2}\right](1/x) = \mathcal{L}^{-1}\left[\frac{1}{s^2}-\frac{\log (s)}{s^2}\right](\log x). $$
The latter can be calculated here.
This leads to
$$ f(x) = \log (x) (\log (\log (x))+\gamma ), $$
where $\gamma$ is the Euler Mascheroni constant.

Edit
Here's a way to compute the inverse Laplace transform:

*

*Integrate by parts

*Introduce a parameter $a = -1$

*Differentiate and integrate w.r.t. $a$ and use the linearity of the transform

*Integrate

*Evaluate the transform

*Differentiate

*Substitute $a = -1$
$$ \begin{align*}
\mathcal{L}_{s}^{-1}\left[\frac{d}{ds} \frac{\log (s)}{s}\right](t) &= -t \, \mathcal{L}_{s}^{-1}\left[\frac{\log (s)}{s}\right](t) \\
&= -t \, \mathcal{L}_{s}^{-1}\left[\log (s) \, s^a\right](t) \\
&= -t \, \frac{d}{da} \mathcal{L}_{s}^{-1}\left[\int \log (s) \, s^a da\right](t) \\
&= -t \, \frac{d}{da} \mathcal{L}_{s}^{-1}\left[s^a\right](t) \\
&= -t \, \frac{d}{da} \frac{t^{-a-1}}{\Gamma(-a)} \\
&= \frac{t^{-a} (\Gamma (-a) \log (t)-\Gamma'(-a))}{\Gamma (-a)^2} \\
&= t (\log (t) - \Gamma'(1)) = t (\log (t) + \gamma).
\end{align*} $$
Generically (ignoring convergence, etc) we can generalize:
$$ \mathcal{L}_{s}^{-1}[f(s)\log(s)](t) = \left.\frac{d}{da}\right|_{a=0} \mathcal{L}_{s}^{-1}[f(s)s^a](t). $$
