On the asymptotically uniform distribution of the zeros of the partial sums of the Riemann zeta function

For every integer n≥2, let S(n)={z:a(n)≤Rez≤b(n)} be the critical strip where all the zeros of the nth partial sum of the Riemann zeta function, ζn(z)=∑k=1n1kz, are located. This paper shows that there exists N such that for n>N the set {Rez:ζn(z)=0} is dense in the interval [a(n),b(n)]. That means that every ζn(z) possesses zeros near every vertical line contained in S(n), provided that n>N.