Global partial-fraction expansion of a Dirichlet L-function logarithmic derivative ($L^{\prime }/L=B_{\chi }-\frac{1}{2}\log (q/\pi )-\frac{1}{2}{\Gamma }^{\prime }/\Gamma ((s+a)/2)+{\sum }_{\rho }((s-\rho {)}^{-1}+{\rho }^{-1})$)

For a primitive nonprincipal Dirichlet character of conductor $q$ and parity $a$, the completed Dirichlet L-function is entire of order one. Its genus-one Hadamard factorization gives the displayed logarithmic derivative, with $B_{\chi }$ the logarithmic derivative of the completion at zero and zeros counted with multiplicity. Combined summands converge locally normally away from zeros. For an imprimitive character induced by ${\chi }^{\ast }$, add ${\sum }_{p\mid Q,p\nmid q}{\chi }^{\ast }(p)\log p/(p^s-{\chi }^{\ast }(p))$ for the removed Euler factors. The primitive principal case is the global partial-fraction expansion of the zeta logarithmic derivative.

// Target: algebraic-number-theory.bigb