Here $(f\cdot w)(s):=f(s)*w(s)$ where $*$ is the group operation.

E.g. $\langle f\rangle_{[a,b]}:=\dfrac{\int_a^bf(x)\,{\mathrm dx}}{b-a}$

where $[a,b]\subseteq{\mathbb R}$ and $w(x):=1$.

For $z\in(0,1)$, we find

$\sum_{k=0}^\infty \langle q\mapsto q\,z^q\rangle_{[k,k+1]}=\dfrac{1}{\ln(z)^2}$,

i.e. (see Natural logarithm of complex numbers)

$\sum_{k=0}^\infty \left(k\,z^k-\langle q\mapsto q\,z^q\rangle_{[k,k+1]}\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$

See Minus twelve . Note.

Function integral, Classical probability density function