The number of zeros of $L'(s,\chi )$
Tom 190 / 2019
Acta Arithmetica 190 (2019), 127-138
MSC: Primary 11M06; Secondary 11M26.
DOI: 10.4064/aa180219-28-9
Opublikowany online: 13 June 2019
Streszczenie
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.
