On the sum of positive divisors functions

Properties of divisor functions σ k ( n ) , defined as sums of k -th powers of all divisors of n , are studied through the analysis of Ramanujan’s differential equations. This system of three differential equations is singular at x = 0 . Solution techniques suitable to tackle this singularity are de...

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Bibliographic Details
Published in:Research in number theory 2021-06, Vol.7 (2), Article 25
Main Authors: Erban, Radek, Van Gorder, Robert A.
Format: Article
Language:English
Subjects:
Differential equations
Dynamical systems
Mathematical analysis
Mathematical functions
Mathematics
Mathematics and Statistics
Number Theory
Singularity (mathematics)
Systems analysis
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Summary:Properties of divisor functions σ k ( n ) , defined as sums of k -th powers of all divisors of n , are studied through the analysis of Ramanujan’s differential equations. This system of three differential equations is singular at x = 0 . Solution techniques suitable to tackle this singularity are developed and the problem is transformed into an analysis of a dynamical system. Number theoretical consequences of the presented dynamical system analysis are then discussed, including recursive formulas for divisor functions.
ISSN:2522-0160
2363-9555
DOI:10.1007/s40993-021-00240-6