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We study a bijective map from integer partitions to the prime factorizations of integers that we call the supernorm of a partition, in which the multiplicities of the parts of partitions are mapped to the multiplicities of prime factors of natural numbers. The supernorm is connected to a family of maps we define, which suggests the potential to apply techniques from partition theory to identify and prove multiplicative properties of integers. We make a brief study of pertinent analytic aspects of the supernorm. Then, as an application of supernorma mappings (i.e., pertaining to the supernorm statistic), we prove an analogue of a formula of Kural-McDonald-Sah to give arithmetic densities of subsets of $mathbb N$ instead of natural densities in $mathbb P$ like previous formulas of this type; this builds on works of Alladi, Ono, Wagner, and the first and third authors. Finally, using a table of supernormal additive-multiplicative correspondences, we conjecture Abelian-type formulas that specialize to our main theorem and other known results.
One of the most basic results concerning the number-theoretic properties of the partition function $p(n)$ is that $p(n)$ takes each value of parity infinitely often. This statement was first proved by Kolberg in 1959, and it was strengthened by Subba
Using a theorem of Frobenius filtered through partition generating function techniques, we prove partition-theoretic and $q$-series Abelian theorems, yielding analogues of Abels convergence theorem for complex power series, and related formulas. As a
In this article we study the norm of an integer partition, which we define to be the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of current rese
We examine partition zeta functions analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over
In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formula for the s