Classical zero-free region for Dirichlet L-functions

There is an absolute $c>0$ such that the Dirichlet L-functions modulo $q$ have no zero inapart from at most one real simple zero belonging to a real nonprincipal character.

For primitive real nonprincipal Dirichlet characters, there is an absolute $c>0$ with at most one zero in this interval, counted with multiplicity. Positivity of $-{\zeta }^{\prime }/\zeta -L^{\prime }/L$ and the local partial fractions give $0\le 1/(\sigma -1)+C\log q-\sum (\sigma -\beta {)}^{-1}$. Testing $\sigma =1+a/\log q$ with small fixed $a$ rules out two nearby real zeros. Any possible zero is consequently simple; existence is not asserted.

An exceptional zero is the possible real simple zero $\beta$ omitted from the otherwise uniform classical zero-free region for Dirichlet L-functions. It belongs to a real nonprincipal primitive Dirichlet character and satisfies $\beta <1$.

For two real primitive nonprincipal Dirichlet characters with nonprincipal product, the real zeros in this interval are counted with multiplicity. The logarithmic-derivative coefficients of $\zeta L({\chi }_1)L({\chi }_2)L({\chi }_1{\chi }_2)$ are $\Lambda (n)(1+{\chi }_1(n))(1+{\chi }_2(n))\ge 0$. The global partial-fraction expansion of a Dirichlet L-function logarithmic derivative therefore gives $0\le 1/(\sigma -1)-{\sum }_{j=1}^2(\sigma -{\beta }_j{)}^{-1}+C\log (q_1{q}_2)$ if two nearby real zeros exist. Put $\sigma =1+a/\log (q_1{q}_2)$, choose $a$ small, and test $1-{\beta }_j<a/(4\log (q_1{q}_2))$. The right side is negative, a contradiction.

// Target: analytic-number-theory.bigb

Fix the exceptional-zero threshold $\beta >1-\eta /\log q$, with $\eta$ less than one third of the constant in Landau theorem for two real Dirichlet characters. If distinct primitive conductors satisfy $q_1<q_2\le q_1^2$, then $\log (q_1{q}_2)\le 3\log q_1$; both exceptional zeros lie in the forbidden two-zero interval. Thus the conductors grow at least quadratically. The same argument with equal conductors proves at most one exceptional real primitive character for each modulus. Existence is not asserted, and a consistently sufficiently small threshold is essential.

// Target: number-theory.bigb

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

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 $ϵ$.