On the zeros of Dirichlet 𝐿-functions. I

Abstract

A mean value theorem for arg ⁡ L ( 1 / 2 + i ( t + h ) , χ ) − arg ⁡ L ( 1 / 2 + i t , χ ) \arg L(1/2 + i(t + h), \chi ) - \arg L(1/2 + it, \chi ) is established. This yields mean estimates for the number of zeros of L ( s , χ ) L(s, \chi ) in small boxes.