Prime number theorem in an arithmetic progression with an exceptional zero

Uniformly for $x\ge 2$, positive integer modulus $q$, and $\gcd (a,q)=1$,where the second term occurs only when a real character ${\chi }_1$ has an exceptional zero $\beta$.

The term involving ${\chi }_1$ is omitted if there is no exceptional zero. Using $\log (2q)$ keeps the error meaningful at $q=1$, where the statement reduces to the ordinary Prime number theorem error rather than falsely asserting an exact equality.

If some $ϵ>0$ satisfiesuniformly for sufficiently large $q$, then every exceptional zero modulo $q$ satisfiesChoose ${\chi }_1(a)=-1$ and $x=\exp (A(\log q{)}^2)$ in the exceptional-zero asymptotic, with $A$ large in terms of $ϵ$.