An estimate of the lower bound of the real parts of the zeros of the partial sums of the Riemann zeta function

Abstract

Let View the MathML source ζn(z):=∑k=1n1kz, z=x+iy, be the n th partial sum of the Riemann zeta function and aζn(z):=inf⁡{ℜz:ζn(z)=0}. In this paper we prove that View the MathML source aζn(z)=−log⁡2log⁡(n−1n−2)+Δn, n>2, with lim⁡supn→∞ |Δn|≤log⁡2