= Landau theorem for two real Dirichlet characters
{c}
{title2=$L(s,\chi_1)L(s,\chi_2)\text{ has at most one real zero }>1-c/\log(q_1q_2)$}
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 $\Lambda(n)(1+\chi_1(n))(1+\chi_2(n))\geq0$. The <global partial-fraction expansion of a Dirichlet L-function logarithmic derivative> therefore gives $0\leq1/(\sigma-1)-\sum_{j=1}^2(\sigma-\beta_j)^{-1}+C\log(q_1q_2)$ if two nearby real zeros exist. Put $\sigma=1+a/\log(q_1q_2)$, choose $a$ small, and test $1-\beta_j<a/(4\log(q_1q_2))$. The right side is negative, a contradiction.
// Target: analytic-number-theory.bigb
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