Is this Euler-Mascheroni constant calculation from double integrals a true identity?

Solution 1:

What justifies diligently the step $\sum\limits_{a=1}^{\infty} \sum\limits_{b=1}^{\infty} x^{a b} \rightarrow \int\limits_{a=1}^{\infty} \int\limits_{b=1}^{\infty} x^{a b}$; this seems likely very vague "what is valid for distances ... valid for areas ... valid for volumes..." so might one argument and extend to higher dimmensions $\int\int\int\int...$ So far cannot follow to justify identity.

Continue that there is no reason to me to believe that this identity works, because if then according to Hadjicostas/Sondow you would have:

$$ \int_0^n \left(\int_k^{\infty } e^{a b} \, da\right) \, db-\int_k^{\infty } \left(\int_0^n e^{a b} \, da\right) \, db=-\int^1_0\int^1_0\frac{1-a}{(1-ab)\log (ab)} da\; db = \gamma$$

isn't it?