A paper of Landau where he uses an integral of $Z(s) := \log \zeta(s)$ against $x^s/s^2$ (not $x^s/s$) was published in 1908 and can be found here. The function $Z(s)$ is introduced on p. 753, a funky contour is drawn on p. 754, and lower down on that page he starts integrating $Z(s)x^s/s^2$ and he converts things into $\pi(x)$ in Section 7.

A paper of Landau where he uses an integral of $Z(s) := \log \zeta(s)$ against $x^s/s^2$ (not $x^s/s$) was published in 1908 and can be found here. The function $Z(s)$ is introduced on p. 753, a funky contour is drawn on p. 754, and lower down on that page he starts integrating $Z(s)x^s/s^2$ and he converts things into $\pi(x)$ in Section 7.

I found this information in Narkiewicz's book "The Development of Prime Number Theory". Section 6.2 is on Landau's approach to PNT and in part 5 (starting on p. 283) he mentions the above paper:

In the year 1908 Landau gave (Landau 1908d) two new proofs of the Prime Number Theorem in its simplest form, i.e. without giving any evaluation of the error term. The first is similar to the proof presented above but instead of dealing with the the function $\zeta'(s)/\zeta(s)$ Landau considers $Z(s) = \log \zeta(s)$. This necessitates a small modification of the integration path to take care of the essential singularity of the integrand at $s=1$ but the remainder of the argument is carried out along the lines of the preceding proof...

The reference Landau 1908d is the paper I link to above.