Abstract and Applied Analysis
Volume 2011 (2011), Article ID 909674, 15 pages
http://dx.doi.org/10.1155/2011/909674
Research Article

The Critical Strips of the Sums 1 + 2 𝑧 + ⋯ + 𝑛 𝑧

G. Mora and J. M. Sepulcre

Department of Mathematical Analysis, University of Alicante, 03080 Alicante, Spain

Received 15 November 2010; Accepted 7 March 2011

Academic Editor: Stephen Clark

Copyright © 2011 G. Mora and J. M. Sepulcre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a partition of the critical strip, associated with each partial sum 1 + 2 𝑧 + ⋯ + 𝑛 𝑧 of the Riemann zeta function for Re 𝑧 < − 1 , formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a large class of almost-periodic functions with bounded spectrum.