Distance between consecutive elements of the multiplicative group of integers modulo $n$

Abstract

For a prime number $p$, we consider its primorial $P:=p\#$ and $U(P):={\left(\mathbb{Z}/P\mathbb{Z}\right)}^\times$ the set of elements of the multiplicative group of integers modulo $P$ which we represent as points anticlockwise on a circle of perimeter $P$. These points considered with wrap around modulo $P$ are those not marked by the Eratosthenes sieve algorithm applied to all primes less than or equal to $p$. In this paper, we are concerned with providing formulas to count the number of gaps of a given even length $D$ in $U(P)$ which we note $K(D,P)$.