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Abstract

Assuming the Generalized Riemann Hypothesis, we show that for $q \gt 1$ and $T \gt 2$, $$ N_1(T,\chi) = \frac{T}{\pi}\log \frac{qT}{2m\pi e} + O\biggl(\frac{\log qT}{\log\log qT} + \sqrt{m\log 2m\cdot\log qT}\bigg), $$ where $N_1(T,\chi)$ is the number of zeros of $L’(s,\chi)$ in the region $\Re s \gt 0$, $|\Im s|\le T$, $\chi$ is a primitive character to the modulus $q$, $m$ is the smallest prime number not dividing $q$, and the implied constant is absolute.