Character-twisted Eisenstein series ($G_k(\chi ,z)$)

For an integer $k>2$ and a Dirichlet character $\chi$ modulo $N\ge 1$ with $\chi (-1)=(-1{)}^k$, defineThe zero pair is explicitly excluded, including when $N=1$. Comparison with a convergent lattice Eisenstein sum proves absolute and locally uniform convergence on the complex upper half-plane, hence holomorphy. Replacing $z$ by $z+N$ permutes the terms by $n\mapsto n+mN$ without changing $\chi (n)$ or its coprimality condition, so the period is $N$. The parity condition makes the terms indexed by $(m,n)$ and $(-m,-n)$ agree instead of cancel.

With $S_{\chi }(r)$ the finite Fourier transform in Gauss sum of a Dirichlet character, the constant coefficient of $G_k(\chi ,z)$ in $e^{2\pi iz/N}$ is $2L(\chi ,k)$, and its positive coefficients areFor a primitive Dirichlet character this simplifies to $2(-2\pi i{)}^{k}g(\chi )((k-1)!N^k{)}^{-1}{\sum }_{r\mid n}\overline{\chi }(r)r^{k-1}$. Without primitivity the simplified formula is generally false.