Privileged Regions in Critical Strips of Non-lattice Dirichlet Polynomials

This paper shows, by means of Kronecker’s theorem, the existence of infinitely many privileged regions called $r$ -rectangles (rectangles with two semicircles of small radius $r$ ) in the critical strip of each function $L_{n}(z)\!:=\!$ $1-\sum _{k=2}^{n}k^{z}$ , $n\!\ge \!2$ , containing exactly $\left \dfrac{T\log n}{2\pi }\right] +1$ zeros of $L_{n}(z)$ , where $T$ is the height of the $r$ -rectangle and $\left\cdot \right]$ represents the integer part.