Distribution in coprime residue classes of polynomially-defined multiplicative functions

An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue classes of polynomially-defined multiplicative functions, establishing equidistribution results allowing a wide range of uniformity in the modulus $q$. For example, we show that the values $\phi(n)$, sampled over integers $n \le x$ with $\phi(n)$ coprime to $q$, are asymptotically equidistributed among the coprime classes modulo $q$, uniformly for moduli $q$ coprime to $6$ that are bounded by a fixed power of $\log{x}$.