# A question about the setup of zero density estimates for Dirichlet $L$-functions

For $$L(s,\chi)= \sum_{n \geq 1}\frac{\chi(n)}{n^s}$$, where $$s = \sigma + it$$, we define the function $$N(\sigma, T, \chi)$$ which counts the zeros $$\rho = \beta + i\gamma$$ for which $$L(\rho, \chi) =0$$ where $$\sigma \leq \beta$$, $$\lvert \gamma\rvert \leq T$$. We know that the upper bounds of $$\sum_{\chi}N(\sigma, T, \chi)$$, are called as the zero-density estimates for $$L(s, \chi)$$. Well, several results exist in literature for such estimates, both asymptotic and explicit. For example, Chen showed that for $$q\geq 3$$, $$\frac{1}{2}\leq \sigma <1$$, $$T \geq \max(\frac{10^5}{q}, 10^4 \log q)$$, $$\sum_{\chi} N(\sigma, T, \chi) \leq \Bigl( \frac{25039}{\log(qT)} + 5700\Bigr)(q^3 T^4)^{1-\sigma} (\log(qT))^{6\sigma}$$, where $$\chi$$ runs through all non-principal characters $$\mod q$$, and subsequently other such explicit results also exist.

My question is whether or not strict inequalities between $$\frac{1}{2}$$ and $$\sigma$$ or between $$\sigma$$ and $$\beta$$ would make any difference to the results achieved for $$\sum_{\chi} N(\sigma, T, \chi)$$. Is it that under the GRH, it makes a theoretical difference but that might not show up in the estimation results used to obtain the explicit results as above?

0. You probably mean that $$\chi$$ runs through primitive Dirichlet characters modulo $$q$$.
1. Changing $$1/2\leq\sigma$$ to $$1/2<\sigma$$ would not make any difference. Indeed, for $$\sigma=1/2$$ better results are known as $$N(1/2,T,\chi)$$ does not exceed the number of all zeros of $$L(s,\chi)$$ in the box $$\{\beta+i\gamma:\text{0\leq\beta\leq 1 and |\gamma|\leq T}\},$$ and we can count the number of all zeros in that box asymptotically (using the functional equation).
2. Changing $$\sigma\leq\beta$$ to $$\sigma<\beta$$ would not make any difference. Indeed, if we define $$N'(\sigma,T,\chi)$$ like $$N(\sigma,T,\chi)$$ but with $$\sigma<\beta$$, then for any $$\sigma>1/2$$ we have $$\sum_\chi N'(\sigma,T,\chi)\leq \sum_\chi N(\sigma,T,\chi)\leq\inf_{\sigma'<\sigma} \sum_\chi N'(\sigma',T,\chi).$$ So the stated bound for $$N(\sigma,T,\chi)$$ yields the same bound for $$N'(\sigma,T,\chi)$$ and vice versa.