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

This paper aims to introduce a new class of entire functions whose zeros (z_k)_k≥1 satisfy ∑_k=1^∞Im z_k=O(1).

This is done by means of a Ritt's formula which is used to prove that every partial sum of the Riemann Zeta function, ζ_n(z):=∑_k=1ⁿ1/k^z, n≥2, has zeros (s_nk)_k≥1 verifying ∑_k=1^∞Re s_nk=O(1) and extending this property to a large class of entire functions denoted by A_O.

It is found that this new class A_O has a part in common with the class A introduced by Levin but is distinct from it. It is shown that, in particular, A_O contains every partial sum of the Riemann Zeta function ζ_n(iz) and every finite truncation of the alternating Dirichlet series expansion of the Riemann zeta function, T_n(iz):=∑_k=1ⁿ(−1)^k−1/k^iz, for all n≥2.

With the exception of the n=2 case, numerical experiences show that all zeros of ζ_n(z) and T_n(z) are not symmetrically distributed around the imaginary axis. However, the fact consisting of every function ζ_n(iz) and T_n(iz) to be in the class A_O implies 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 of ζ_n(z) and T_n(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.

The 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.