The asymptotic number of zeros of exponential sums in critical strips

TL;DR

This paper investigates the asymptotic distribution of zeros of normalized exponential sums within critical strips, extending classical results and employing Backlund's lemma to analyze zeros in individual strips.

Contribution

It derives the asymptotic count of zeros in each vertical strip for exponential sums, building on and extending prior work by Langer and Moreno.

Findings

Asymptotic zero count in individual strips established

Almost every vertical line intersects finitely many zeros

Extension of classical zero distribution results

Abstract

Normalized exponential sums are entire functions of the form $$ f(z)=1+H_1e^{w_1z}+\cdots+H_ne^{w_nz}, $$ where $H_1,\ldots, H_n\in\C$ and $0<w_1<\ldots<w_n$. It is known that the zeros of such functions are in finitely many vertical strips $S$. The asymptotic number of the zeros in the union of all these strips was found by R. E. Langer already in 1931. In 1973, C. J. Moreno proved that there are zeros arbitrarily close to any vertical line in any strip $S$, provided that $1,w_1,\ldots,w_n$ are linearly independent over the rational numbers. In this study the asymptotic number of zeros in each individual vertical strip is found by relying on R. J. Backlund's lemma, which was originally used to study the zeros of the Riemann $\zeta$-function. As a counterpart to Moreno's result, it is shown that almost every vertical line meets at most finitely many small discs around the zeros of $f$.