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

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