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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.
ISSN 0022-314X

**Full text not available from this repository.**

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 |
---|---|

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of 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 |

Date: | April 2011 |

Volume: | Volume 131 |

Number: | Number 4 |

Page Range: | pp. 745-749 |

Identification Number: | 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) |

References: | [1] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997) 235–265, see also http://magma.maths.usyd.edu.au/magma/. [2] P. Erdo˝ s, Problem 4493, Amer. Math. Monthly 59 (1952) 412; Solution J.B. Kelly, Amer. Math. Monthly 60 (1953) 557–558. [3] P. Erdo˝ s, M. Kac, Problem 4518, Amer. Math. Monthly 60 (1953) 47; Solution R. Breusch, Amer. Math. Monthly 61 (1954) 264–265. [4] J.B. Friedlander, F. Luca, M. Stoiciu, On the irrationality of a divisor function series, Integers 7 (A31) (2007), 9 pp. (electronic). [5] R.K. Guy, Unsolved Problems in Number Theory, third edition, Springer-Verlag, 2004. [6] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, fifth edition, Oxford University Press, 1979. [7] A. Schinzel, W. Sierpin´ ski, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. IV (1958) 185–208, Erratum: Acta Arith. V (1958) 259. [8] J.-C. Schlage-Puchta, The irrationality of some number theoretical series, Acta Arith. 126 (4) (2007) 295–303. |

URI: | http://wrap.warwick.ac.uk/id/eprint/41557 |

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