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On the Q-linear independence of the sums ∑n=1∞σk(n)/n!
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Deajim, Abdulaziz and Siksek, Samir (2011) On the Q-linear independence of the sums ∑n=1∞σk(n)/n! Journal of Number Theory, Volume 131 (Number 4). pp. 745-749. doi:10.1016/j.jnt.2010.11.009 ISSN 0022-314X.
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Official URL: http://dx.doi.org/10.1016/j.jnt.2010.11.009
Abstract
Let sigma(k)(n) denote the sum of the k-th powers of the positive divisors of n. Erdos and Kac conjectured that the sum
alpha(k) = Sigma(infinity)(n=1) sigma(k)(n)/n!
is irrational for k >= 1. This is known to be true for k = 1, 2 and 3. Fix r >= 1. In this article we give a precise criterion for 1, alpha(1), ..., alpha(r) to be Q-linearly independent, assuming a standard conjecture of Schinzel on the prime values taken by a family of polynomials. We have verified our criterion for r = 50. (C) 2011 Elsevier Inc. All rights reserved.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Linear dependence (Mathematics) | ||||
Journal or Publication Title: | Journal of Number Theory | ||||
Publisher: | Academic Press | ||||
ISSN: | 0022-314X | ||||
Official Date: | April 2011 | ||||
Dates: |
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Volume: | Volume 131 | ||||
Number: | Number 4 | ||||
Page Range: | pp. 745-749 | ||||
DOI: | 10.1016/j.jnt.2010.11.009 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC) |
Data sourced from Thomson Reuters' Web of Knowledge
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