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

Abstract

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