Comparing zeros of distinct Dirichlet L-functions

Abstract

For any $\hat{I}¸>\frac{1}{3}$, we show that there are constants $c_1,c_2>0$ that depend only on $\hat{I}¸$ for which the following property holds. If $\ddot{I}{‡}_1,\ddot{I}{‡}_2$ are two distinct primitive Dirichlet characters modulo $q$, and $T\ge c_1{q}^{\hat{I}}¸$, then $L(s,\ddot{I}{‡}_1)$ and $L(s,\ddot{I}{‡}_2)$ do not have the same zeros in the region $$\{s=\ddot{I}ƒ+it\in \mathbb{C}:0<\ddot{I}ƒ<1,T<t<T+c_2{q}^{\hat{I}}¸\log T\}.$$ For cubefree moduli $q$, the same result holds for any $\hat{I}¸>\frac{1}{4}$.

7 pages; v.2 improves the main theorem stated in v.1 and implements various suggestions of the referees