2014/12/09 by Colin Defant, Defant, Colin · 1 citation
Mathematics · #11N80 #11R11 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #math.NT #msc:11N80 #msc:11R11
paper · pdf · doi:10.48550/arxiv.1412.3105
14 pages, 0 figures, Supported by National Science Foundation grant no. 1262930. arXiv admin note: text overlap with arXiv:1412.3072
arxiv created 2014/12/09 · openalex publication_date 2014/12/09 · arxiv updated 2014/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A unitary divisor c of a positive integer n is a positive divisor of n that is relatively prime to (n)/(c). For any integer k, the function σk^* is a multiplicative arithmetic function defined so that σk^*(n) is the sum of the kth powers of the unitary divisors of n. We provide analogues of the functions σk^* in imaginary quadratic rings that are unique factorization domains. We then explore properties of what we call n-powerfully unitarily t-perfect numbers, analogues of the unitary multiperfect numbers that have been defined and studied in the integers. We end with a list of several opportunities for further research.