Elementary Question on Dirichlet L-functions
 

Given [TEX]\epsilon > 0[/TEX], [TEX]K(\epsilon) \ge 2[/TEX], and [TEX]c > 0[/TEX], such that if [TEX]q > K[/TEX] and for every [TEX]d \mid q[/TEX] with [TEX]\chi[/TEX] a primitive character [TEX]\pmod d[/TEX] we have if [TEX]L(s, \chi) \ne 0[/TEX], for [TEX]Re(s) > 1 - \frac{1}{(\log q)^{3/4}}[/TEX] and [TEX]|t| \le \exp(\epsilon \left(\log q \right)^{3/4})[/TEX], then, for any [TEX]a[/TEX] with [TEX]\gcd(a,q) = 1[/TEX], we have [TEX]\pi(x;q,a) \ge \frac{cx}{\varphi(q)\log x}[/TEX] whenever [TEX]q < x^{0.472}[/TEX]. This is Theorem 2 of Harman's paper (which I have attached to this post).

Then does this mean that the nontrivial zeros of [TEX]L(s,\chi)[/TEX] lie on the line [TEX]Re(s) = \frac{1}{(\log q)^{3/4}}[/TEX]?

[QUOTE=intrigued;253533]Given [TEX]\epsilon > 0[/TEX], [TEX]K(\epsilon) \ge 2[/TEX], and [TEX]c > 0[/TEX], such that if [TEX]q > K[/TEX] and for every [TEX]d \mid q[/TEX] with [TEX]\chi[/TEX] a primitive character [TEX]\pmod d[/TEX] we have if [TEX]L(s, \chi) \ne 0[/TEX], for [TEX]Re(s) > 1 - \frac{1}{(\log q)^{3/4}}[/TEX] and [TEX]|t| \le \exp(\epsilon \left(\log q \right)^{3/4})[/TEX], then, for any [TEX]a[/TEX] with [TEX]\gcd(a,q) = 1[/TEX], we have [TEX]\pi(x;q,a) \ge \frac{cx}{\varphi(q)\log x}[/TEX] whenever [TEX]q < x^{0.472}[/TEX]. This is Theorem 2 of Harman's paper (which I have attached to this post).

Then does this mean that the nontrivial zeros of [TEX]L(s,\chi)[/TEX] lie on the line [TEX]Re(s) = \frac{1}{(\log q)^{3/4}}[/TEX]?[/QUOTE]

Dirichlet L-functions are believed to follow GRH, so the answer would be no.