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Classical zero-free region for Dirichlet L-functions

Siegel zero

Prime number theorem in an arithmetic progression with an exceptional zero

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 $\epsilon$.


Exceptional-zero bound from a uniform upper bound in arithmetic progressions

Exceptional-zero bound from a uniform upper bound in arithmetic progressions

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Exceptional-zero bound from a uniform upper bound in arithmetic progressions

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