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About the Cover: Lambert Series

About the Cover: Lambert Series Comput. Methods Funct. Theory (2017) 17:1–2 DOI 10.1007/s40315-017-0193-3 Elias Wegert © Springer-Verlag Berlin Heidelberg 2017 The image on the front page shows a (modified) phase plot of the 10th partial sum s of the Lambert series: f (z) = , |z| < 1. 1 − z n=0 This series converges in the unit disk and its sum is the generating function of the divisor function, i.e., the coefficients d of the Taylor series f (z) = d z n n n=0 coincide with the numbers of positive integral divisors of n.For |z| > 1, the partial sums s (z) tend to ∞, while their phases s /|s | converge to −1, which explains the n n n almost uniform color in the exterior of the unit circle on the cover image. 123 2 E. Wegert To adapt the appearance of the image to the layout of the CMFT journal, the function has been multiplied by a unimodular constant, which converts the dominating light blue associated with negative real values to yellow. The figure above depicts a phase plot of f (left) and its modication with the same color shift (right) in the unit disk. Divergence of the series in http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

About the Cover: Lambert Series

Computational Methods and Function Theory , Volume 17 (1) – Feb 20, 2017

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-017-0193-3
Publisher site
See Article on Publisher Site

Abstract

Comput. Methods Funct. Theory (2017) 17:1–2 DOI 10.1007/s40315-017-0193-3 Elias Wegert © Springer-Verlag Berlin Heidelberg 2017 The image on the front page shows a (modified) phase plot of the 10th partial sum s of the Lambert series: f (z) = , |z| < 1. 1 − z n=0 This series converges in the unit disk and its sum is the generating function of the divisor function, i.e., the coefficients d of the Taylor series f (z) = d z n n n=0 coincide with the numbers of positive integral divisors of n.For |z| > 1, the partial sums s (z) tend to ∞, while their phases s /|s | converge to −1, which explains the n n n almost uniform color in the exterior of the unit circle on the cover image. 123 2 E. Wegert To adapt the appearance of the image to the layout of the CMFT journal, the function has been multiplied by a unimodular constant, which converts the dominating light blue associated with negative real values to yellow. The figure above depicts a phase plot of f (left) and its modication with the same color shift (right) in the unit disk. Divergence of the series in

Journal

Computational Methods and Function TheorySpringer Journals

Published: Feb 20, 2017

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