Journal title
Research in Number Theory
DOI
10.1007/s40993-021-00240-6
Volume
7
Last updated
2024-09-05T08:30:14.693+01:00
Abstract
<p>Properties of divisor functions <span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C3;</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></math>">σk(n)σk(n)</span>, defined as sums of <em>k</em>-th powers of all divisors of <em>n</em>, are studied through the analysis of Ramanujan’s differential equations. This system of three differential equations is singular at <span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>">x=0x=0</span>. 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.</p>
Symplectic ID
1161323
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Publication type
Journal Article
Publication date
22 Mar 2021