Exceptional-zero bound from a uniform upper bound in arithmetic progressions
= Exceptional-zero bound from a uniform upper bound in arithmetic progressions
If some $\epsilon>0$ satisfies
$$
\psi(x;q,a)\leq(2-\epsilon)\frac{x}{\varphi(q)}
\qquad(x\geq q^2)
$$
uniformly for sufficiently large $q$, then every <exceptional zero> modulo $q$ satisfies
$$
\beta\leq1-\frac{c_\epsilon}{(\log q)^2}.
$$
Choose $\chi_1(a)=-1$ and $x=\exp(A(\log q)^2)$ in the exceptional-zero asymptotic, with $A$ large in terms of $\epsilon$.