2024/01/08 by A. Nader, Nader, Alfredo
Mathematics · #05A15 (Secondary) #05A17 (Primary) 11P81 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2401.04026
openalex publication_date 2024/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.