 Landau theorem for two real Dirichlet characters

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