Function whose power series coefficients contain logarithms

Question

Is there a function that can be expressed as a power series

$$f(x)=\sum_{n=0}^\infty a_n x^n$$

whose coefficients $a_n$ are expressions containing $\log n$ or something similar?

Answer 1

Here is a non-elementary example arising from zeta functions.

Consider the polylogarithm $\text{Li}_s(x)=\sum_{n=1}^\infty \dfrac{x^n}{n^s}$. Since $n^{-s}=\exp(-s \ln n)$, differentiation with respect to $s$ yields $\partial_s\text{Li}_s(x)=-\sum_{n=1}^\infty \ln n\,\dfrac{x^n}{n^s}$.

Note in particular that $x\to 1$ gives the first derivative of the Riemann zeta function; the limit $s\to 0$ thus yields $\zeta'(0)=-\sum_{n=1}^\infty \ln n$ (in the sense of analytic continuation in complex $s$.