Zeros of Partial Sums of the Riemann Zeta-function

We write ρX = βX + iγX for a typical zero of FX(s). The number of these up to height T we denote by NX(T ), and the number of these with βX ≥ σ by NX(σ, T ). We follow the convention that if T is the ordinate of a zero, then NX(T ), say, is defined as limǫ→0+ NX(T + ǫ). There are two natural ways to pose questions about NX(T ), NX(σ, T ), and the distribution of the zeros generally. We can fix an X and consider zeros with 0 < γX ≤ T and let T tend to infinity, or we can ask for results that are uniform as X and T both tend to infinity. Here we will be concerned with the latter sort of question.