I don't know if this approach was in the literature, I would like to know some expression for $\Re s>1$ of the product $$\zeta(s)b(s),$$ where $\zeta(s)$ is the Riemann Zeta function, and $b(s)$ is defined in this section 9. Integral representation and continuation of the Wikipedia article concerning to Bernoulli Numbers.
Using Riemann's trick, I say the change of variables, $t=nx$ for integers $n\geq 1$ if it is possible to justify, one has $$b(s)\zeta(s)=2s e^{\frac{\pi i s}{2}}\int_{0}^\infty \left(\sum_{n=1}^\infty \frac{1}{1-e^{2\pi n x}}\right)x^{s-1} dx.$$
I know that LHS converges for $\Re s>1$, thus by absolute convergence one could to show also that RHS converges. Also I know that it's possible to get, using Wolfram Alpha, a closed form for the series in the integrand (but after I don't know what summands I can integrate separately). See if you need this code in Wolfram Language
sum 1/(1-e^(2 pi n x)), from n=1 to infinite
the condition for the convergence is satisfied since $e^{\pi \Re x}>0$ holds.
Question. For $\Re s>1$ can you get a closed-form, following my attempt or yours, explaining me the convergence issues, for the product $$\zeta(s)b(s)?$$ If you known from the literature some remarkable identity you can refers it. My attempt was this, since seems to me the more simple, and since I believe that could be interesting to get applications. Thus, can you justify my approach or get a different one? Thanks in advance.
Notice also that with the online calculator one has for $\Re s>1$ that
integrate t^s/(1-e^{2 pi t})dt/t, from t=0 to infinite.