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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.

This paper aims to present a new method for obtaining points of the set determined by the closure of the real projections of the zeros of each partial sum 1+2^s+ċ+n^s, n≥2, s=σ+it, of the Riemann zeta function and to show several applications of this result.

The authors utilize an auxiliary function related to a known result of Avellar that characterizes the set of points of interest. Several figures and numerical experiences are presented to illustrate the various properties which are studied.

It is first shown that each point of the image of the auxiliary function can be approximated by points of the image of the function formed by the approximants. Secondly, conditions are given on the auxiliary function to obtain points satisfying the property of density which is studied. Finally, by using these conditions, several useful applications are presented to the case n=4 and σ₀=0 which a more specific criterion is also given.

This research is applicable for finding accumulation points of the set of the real projection of the zeros of the approximants on its critical interval. An exact interval included in this set is given for the case n=4. Also, it is demonstrated that the point 0 is included for a large set of values of n.

The method employed is original and it contributes to the study on the properties of the density of the real parts of the zeros of a particular class of entire functions.

Measurement of diminishing or divergent cross section dispersion in a panel plays an important role in the assessment of convergence or divergence over time in key economic indicators. Econometric methods, known as weak σ-convergence tests, have recently been developed (Kong, Phillips, & Sul, 2019) to evaluate such trends in dispersion in panel data using simple linear trend regressions. To achieve generality in applications, these tests rely on heteroskedastic and autocorrelation consistent (HAC) variance estimates. The present chapter examines the behavior of these convergence tests when heteroskedastic and autocorrelation robust (HAR) variance estimates using fixed-b methods are employed instead of HAC estimates. Asymptotic theory for both HAC and HAR convergence tests is derived and numerical simulations are used to assess performance in null (no convergence) and alternative (convergence) cases. While the use of HAR statistics tends to reduce size distortion, as has been found in earlier analytic and numerical research, use of HAR estimates in nonparametric standardization leads to significant power differences asymptotically, which are reflected in finite sample performance in numerical exercises. The explanation is that weak σ-convergence tests rely on intentionally misspecified linear trend regression formulations of unknown trend decay functions that model convergence behavior rather than regressions with correctly specified trend decay functions. Some new results on the use of HAR inference with trending regressors are derived and an empirical application to assess diminishing variation in US State unemployment rates is included.