Accumulation points of the sets of real parts of zeros of the partial sums of the Riemann zeta function

Abstract

It is shown that, for every integer n>2, there exists \delta_{n}>0 such that the closure of the set of the real parts of the zeros of the nth partial sum of the Riemann zeta function \zeta_{n} contains to the interval [-\delta_{n},b^{n}]. b^{n} is the supremum of the real parts of the zeros of \zeta_{n}. It is also demonstrated that b^{n} is positive for all n>2. It is also shown that 0 is an accumulation point common to all the sets P_{\zeta_{n}} wich are the sets of the real parts of the zeros of \zeta_{n}.