It is known that the original series formula for the Riemann zeta function, sum n=1...inf 1/n^s, converges absolutely for Re(s)>1 and does not converge, even conditionally, for Re(s)=1. It is also trivial that the real-line integral formula for zeta, inf x=0...inf x^(s-1)/(e^x-1)/gamma(s)dx, converges absolutely for Re(s)>1.
However, I could not find any information on whether the aforementioned integral converges for Re(s)=1 (with s!=1, of course) in either direction. I believe that the integral does not converge for Re(s)=1 either, but I could not prove it with my current knowledge.
Is there any proof, in any direction, of convergence, or lack thereof, of the integral $$\int_0^\infty \frac{x^{s-1}}{e^x-1} dx$$ on $\Re(s)=1$?
$$\int_1^\infty \frac{x^{s-1}}{e^x-1}dx$$ converges without problem for any $s$.
For $\Re(s)> 0$ then $$\int_0^1 x^{s-1}(\frac{1}{e^x-1}-\frac1x)dx$$ converges.
So it suffices to investigate the convergence of $$\int_0^1 x^{s-2}dx$$