The Prime Index Function

Abstract

In this paper we introduce the prime index function

$$\iota \left( n \right) = {\left( { - 1} \right)^{\pi \left( n \right)}},$$

where π(n) is the prime counting function. We study some elementary properties and theories associated with the partial sums of this function given by

$$\xi \left( x \right): = \sum\limits_{n \le x} {\iota \left( n \right).} $$