Elementary Question on Dirichlet L-functions

intrigued · 2011-02-23, 20:10

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

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

R.D. Silverman · 2011-02-23, 22:15

Quote:

Originally Posted by intrigued

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

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

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

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