  Let $g_1, \dots , g_M$ be additive functions for which there exist
nonconstant polynomials $G_1, \dots , G_M$ satisfying $g_i(p) = G_i(p)$ for all
primes $p$ and all $i \in \{1, \dots , M\}$. Under fairly general and nearly
optimal hypotheses, we show that the functions $g_1, \dots , g_M$ are jointly
equidistributed among the residue classes to moduli $q$ varying uniformly up to
a fixed but arbitrary power of $\log x$. Thus, we obtain analogues of the
Siegel-Walfisz Theorem for primes in arithmetic progressions, but with primes
replaced by values of such additive functions. Our results partially extend
work of Delange from fixed moduli to varying moduli, and also generalize recent
work done for a single additive function.
