Closed-Form Formula for the Partition Function and Related Functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914634378772480 |
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| author | Nader, Alfredo |
| author_facet | Nader, Alfredo |
| contents | We develop a new closed-form arithmetic and recursive formula for the partition function and a generalization of Andrews' smallest parts (spt) function. Using the inclusion-exclusion principle, we additionally develop a formula for the not-relatively prime partition function (which counts the number of partitions that are not relatively prime). Moreover, we prove a theorem involving the greatest common divisor of partitions, which allows us to link partitions to prime numbers and lets us derive a formula for the relatively prime function. Lastly, we develop numerous new identities for Jordan's totient function of second order, Euler's totient function, and Dedekind's psi function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04026 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Closed-Form Formula for the Partition Function and Related Functions Nader, Alfredo Number Theory Combinatorics 05A17 (Primary) 11P81, 05A15 (Secondary) We develop a new closed-form arithmetic and recursive formula for the partition function and a generalization of Andrews' smallest parts (spt) function. Using the inclusion-exclusion principle, we additionally develop a formula for the not-relatively prime partition function (which counts the number of partitions that are not relatively prime). Moreover, we prove a theorem involving the greatest common divisor of partitions, which allows us to link partitions to prime numbers and lets us derive a formula for the relatively prime function. Lastly, we develop numerous new identities for Jordan's totient function of second order, Euler's totient function, and Dedekind's psi function. |
| title | Closed-Form Formula for the Partition Function and Related Functions |
| topic | Number Theory Combinatorics 05A17 (Primary) 11P81, 05A15 (Secondary) |
| url | https://arxiv.org/abs/2401.04026 |