If $x\to 0$, shouldn't $\log(x)\to -\infty$? Then why $\frac{1}{(\pi/4)^\beta|\log(x)|^\beta}$ is convergent $\forall \beta$ near $x=0$? If $\beta$ is negative, then shouldn't it be divergent?
Also, is $\int\frac{1}{(\pi/4)^\beta|\log(x)|^\beta}$ integrable near $x=0$ $\forall \beta$?