Distribution in coprime residue classes of polynomially-defined multiplicative functions

An integer-valued multiplicative function f is said to be polynomially-defined if there 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 functions, establishing equidistribution results allowing wide range uniformity modulus q. For example, we show that values $$\phi (n)$$ , sampled over integers $$n \le x$$ q, are asymptotically equidistributed among modulo uniformly moduli q 6 bounded by fixed power $$\log {x}$$ .