Level Totients for Integer Mosaics

Abstract

We study a level analog of Euler's totient function for integer mosaics. Let $P_i(n)$ be the set of primes appearing in the first $i$ levels of the mosaic of $n$, and let $φ_i(n)$ count the integers $m\leq n$ for which $P_i(m)\cap P_i(n)=\varnothing$. We prove a Möbius divisor-sum formula for $φ_i(n)$ and reduce it to a sum over a set $\mathcal{V}_{i,S}$ of powerful integers. If $S\neq\varnothing$, $i\geq 2$, and $q=\min S$, then \[ |\mathcal{V}_{i,S}\cap[1,x]|\sim C_{i,S}x^{1/q}, \] with $C_{i,S}$ an explicit positive Euler-product constant. For fixed $S$, the density $δ_i(S)$ of integers whose first $i$ levels avoid $S$ exists and has an Euler product; for nonempty $S$, $i\ge2$, and $q=\min S$, the number of such integers up to $N$ is $δ_i(S)N+O_{i,S}(N^{1/q})$. Taking $S=P_i(n)$ gives \[ \frac{φ_i(n)}{n}=δ_i(P_i(n))+O_\varepsilon(n^{-1/2+\varepsilon}) \] uniformly in $n$.