On the non-isolation of the real projections of the zeros of exponential polynomials

Abstract

This paper proves that the real projection of each zero of any function P(z)P(z) in a large class of exponential polynomials is an interior point of the closure of the set of the real parts of the zeros of P(z)P(z). In particular it is deduced that, for each integer value of n≥17n≥17, if z0=x0+iy0z0=x0+iy0 is an arbitrary zero of the n th partial sum of the Riemann zeta function ζn(z)=∑j=1n1jz, there exist two positive numbers ε1ε1 and ε2ε2 such that any point in the open interval (x0−ε1,x0+ε2)(x0−ε1,x0+ε2) is an accumulation point of the set defined by the real projections of the zeros of ζn(z)ζn(z).