Counting Zeros of Dirichlet $L$-Functions

We give explicit upper and lower bounds for $N(T,χ)$, the number of zeros of a Dirichlet $L$-function with character $χ$ and height at most $T$. Suppose that $χ$ has conductor $q>1$, and that $T\geq 5/7$. If $\ell=\log\frac{q(T+2)}{2π}> 1.567$, then \begin{equation*} \left| N(T,χ) - \left( \frac{T}π \log\frac{qT}{2πe} -\frac{χ(-1)}{4}\right) \right| \le 0.22737 \ell + 2 \log(1+\ell) - 0.5. \end{equation*} We give slightly stronger results for small $q$ and $T$. Along the way, we prove a new bound on $|L(s,χ)|$ for $σ<-1/2$.

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