Zeroes of partial sums of the zeta-function

Abstract

This article considers the positive integers N for which ζN (s) = N n=1 n −s has zeroes in the half-plane (s) > 1. Building on earlier results, we show that there are no zeroes for 1 N 18 and for N = 20, 21, 28. For all other N there are infinitely many such zeroes.

On the real projections of zeros of almost periodic functions

“…Since (15) and (16) imply g(α + τ ) > 0 and g(β − τ ) > 0, respectively, we get a contradiction with (14). Now the result follows.…”

“…The study of the zeros of the class of exponential polynomials of type (1) has become a topic of increasing interest, see for example [2,6,10,12,13,14,16,17,18,19]. In this paper, we will study certain properties on the zeros of an almost periodic function f (s) in its vertical strip of almost periodicity U = {s = σ+it : α < σ < β}.…”

Zeros of partial sums of L-functions

“…In particular, this means that for N sufficiently large, ζ N (s) has zeros in σ > 1. Such results have since been made explicit by the work of Monach [25], van de Lune and te Riele [23], and Spira [35,36], followed by recent work of Platt and Trudgian [31] which completes the classification of the finite number of N 's for which ζ N has no zeroes in σ > 1. Zeros of ζ N (s) have been widely studied and various other results are known in the literature due to a number of mathematicians.…”

"On the real projections of zeros of analytic almost periodic functions"

“…On the other hand, the study of the zeros of the class of exponential polynomials of type (1.1) has become a topic of increasing interest, see for example [2,6,10,11,13,14,15,17,18,19,21]. In this paper, we will study certain properties on the zeros of an analytic almost periodic function f (s) in its vertical strip of almost periodicity U = {s = σ + it : α < σ < β}.…”