Free Distributional Data of Arithmetic Functions and Corresponding Generating Functions Determined by Gaps Between Primes

摘要

In this paper, we study free probability on the algebra A consisting of all arithmetic functions, determined by the gaps between primes. As a continuation and as an application of Cho (Classification on Arithmetic Functions and Free-Moment L -Functions. Submitted to B. Korean Math Soc, 2013), we study moment series and R-transforms induced by both arithmetic functions and gaps between primes. Also, we consider R-transform calculus on the arithmetic algebra.