TITLE: Explicit zero density estimate for Dirichlet $L$-functions
QUESTION [6 upvotes]: Let's define $N(\alpha,T,\chi)=\sharp\lbrace \rho=\sigma+i\gamma: L(\rho,\chi)=0, \alpha\leq \sigma<1, |\gamma|\leq T\rbrace$ , where $\chi$ is a primitive Dirichlet character. We know, from Gallagher's paper, that
$$\sum_{q\leq T}\sum_{\chi\pmod{q}}N(\alpha,T,\chi)\ll T^{c(1-\alpha)}$$
for every $0\leq \alpha<1$ with $c>0$ an absolute costant.
But what about an explicit value of such costant $c$? I need some good and explicit estimate of the sum above. Can anyone help me?

REPLY [5 votes]: Matti Jutila (On Linnik's Constant, Math. Scand. 41 (1977), 45-62) proved that if $Q\geq 2$, $T\geq 1$, and $4/5\leq\alpha\leq 1$, and $\epsilon>0$, then
$\sum_{q\leq Q}\sideset{}{'}\sum_{\chi\bmod{q}}N(\alpha,\chi,T)\ll_{\epsilon}(Q^2 T)^{(2+\epsilon)(1-\alpha)},$
where the decoration ' indicates a sum over primitive characters.  (The restriction to primitive characters is necessary.)  The density hypothesis (which follows from the generalized Riemann hypothesis for Dirichlet $L$-functions) predicts that the above estimate holds for $1/2\leq\alpha\leq 1$, and so we have the density hypothesis in a limited range of $\alpha$.  Montgomery's bound
$\sum_{q\leq Q}\sideset{}{'}\sum_{\chi\bmod q}N(\alpha,T,\chi)\ll(Q^2 T)^{\frac{3(1-\alpha)}{2-\alpha}}(\log QT)^{13},\qquad 1/2\leq\alpha\leq 4/5$
should suffice for most applications when $\alpha$ is far from 1.