The asymptotic number of zeros of exponential sums in critical strips

• Published in: Monatshefte für Mathematik Vol. 194; no. 2; pp. 261 - 273
• Main Authors: Heittokangas, Janne, Wen, Zhi-Tao
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.02.2021; Springer Nature B.V
• Subjects: Asymptotic properties; Entire functions; Mathematics; Mathematics and Statistics; Strip; Sums; 30D20; Zero distribution; Backlund’s lemma; Zero-free region; Critical strip; Exponential sum; 30D35
• ISSN: 0026-9255; 1436-5081
• DOI: 10.1007/s00605-020-01489-2

Normalized exponential sums are entire functions of the form f ( z ) = 1 + H 1 e w 1 z + ⋯ + H n e w n z , where H 1 , … , H n ∈ C and 0 < w 1 < ⋯ < 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 Langer (Bull Am Math Soc 37(4):213–239, 1931). Moreno (Compos Math 2(6):69–78, 1973) proved that there are zeros arbitrarily close to any vertical line in any strip S , provided that 1 , w 1 , … , 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 ζ -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 .