I recently encountered a power series similar to the one of the $\log(1-x)$ of the form
$$ F(x)= \sum_{n=1}^\infty \frac{\psi(n)x^n}{n}, $$ where $\psi$ is some Dirichlet character. Has anyone here seen a function like this? Here are some observations I have made:
1) The radius of convergence is the same as for $\log(1-x)$, so the power series converges for $|x| < 1$.
2) If $\psi(n)$ is the trivial characer mod $N$ then $F(x) = \sum_{n=1}^\infty \frac{x^n}{n}-\frac{x^{Nn}}{Nn}=-\log\left(\frac{1-x}{1-x^N}\right)$.
The next interesting case would be when $\psi$ is a quadratic character. I'd be happy about any reference or further observation on these functions.
Such a function is actually a combination of logarithms. Suppose $\psi$ is a Dirichlet character mod $N$, and let $\zeta$ be a primitive $N$th root of unity. The functions $a \mapsto \zeta^{ka}$, for $k=0, ..., N-1$, form a basis of $L^2(\mathbb Z/N\mathbb Z, \mathbb C)$, hence there exist numbers $b_k \in \mathbb C$ such that
$$\psi(n) = \sum_{k=0}^{N-1} b_k \zeta^{kn}$$
for all $n\in \mathbb Z/N\mathbb Z$. The numbers $b_k$ are essentially Gauss sums - the Fourier coefficients of $\psi$. Then your function is just
$$-\sum_{k=0}^{N-1} b_k \log(1-\zeta^kx).$$
Remark that we don't even need the fact that $\psi$ is a character - it could be any periodic function with a period of $N$, and the "twisted" logarithm would still decompose as above, with different $b_k$'s.
It is an interesting idea, nevertheless. For instance, the value at $x=1$ of your function is $L(\psi, 1)$, a number of significant arithmetic interest.