On the sum of positive divisors functions
<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><...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Springer
2021
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| _version_ | 1826269978140082176 |
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| author | Erban, R Van Gorder, R |
| author_facet | Erban, R Van Gorder, R |
| author_sort | Erban, R |
| collection | OXFORD |
| description | <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> |
| first_indexed | 2024-03-06T21:33:40Z |
| format | Journal article |
| id | oxford-uuid:458414a1-9b33-48e4-8b93-b21277861a33 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:33:40Z |
| publishDate | 2021 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:458414a1-9b33-48e4-8b93-b21277861a332022-03-26T15:08:15ZOn the sum of positive divisors functionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:458414a1-9b33-48e4-8b93-b21277861a33EnglishSymplectic ElementsSpringer2021Erban, RVan Gorder, R<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> |
| spellingShingle | Erban, R Van Gorder, R On the sum of positive divisors functions |
| title | On the sum of positive divisors functions |
| title_full | On the sum of positive divisors functions |
| title_fullStr | On the sum of positive divisors functions |
| title_full_unstemmed | On the sum of positive divisors functions |
| title_short | On the sum of positive divisors functions |
| title_sort | on the sum of positive divisors functions |
| work_keys_str_mv | AT erbanr onthesumofpositivedivisorsfunctions AT vangorderr onthesumofpositivedivisorsfunctions |