A class of functions whose sum of zeros has bounded real part

Abstract

PurposeThis paper aims to introduce a new class of entire functions whose zeros (zk)k≥1satisfy ∑k=1∞Im zk=O(1).Design/methodology/approachThis is done by means of a Ritt's formula which is used to prove that every partial sum of the Riemann Zeta function,ζn(z):=∑k=1n1/kz,n≥2, has zeros (snk)k≥1verifying ∑k=1∞Re snk=O(1) and extending this property to a large class of entire functions denoted byAO.FindingsIt is found that this new classAOhas a part in common with the classAintroduced by Levin but is distinct from it. It is shown that, in particular,AOcontains every partial sum of the Riemann Zeta functionζn(iz) and every finite truncation of the alternating Dirichlet series expansion of the Riemann zeta function,Tn(iz):=∑k=1n(−1)k−1/kiz, for alln≥2.Practical implicationsWith the exception of then=2 case, numerical experiences show that all zeros ofζn(z) andTn(z) are not symmetrically distributed around the imaginary axis. However, the fact consisting of every functionζn(iz) andTn(iz) to be in the classAOimplies the existence of a very precise physical equilibrium between the zeros situated on the left half‐plane and the zeros situated on the right half‐plane of each function. This is a relevant fact and it points out that there is certain internal rule that distributes the zeros ofζn(z) andTn(z) in such a way that few zeros on the left of the imaginary axis and far away from it, must be compensated with a lot of zeros on the right of the imaginary axis and close to it, and vice versa.Originality/valueThe paper presents an original class of entire functions that provides a new point of view to study the approximants and the alternating Dirichlet truncations of the Riemann zeta function.