  We consider the distribution in residue classes modulo primes $p$ of Euler's
totient function $\phi(n)$ and the sum-of-proper-divisors function
$s(n):=\sigma(n)-n$. We prove that the values $\phi(n)$, for $n\le x$, that are
coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime
residue classes modulo $p$, uniformly for $5 \le p \le (\log{x})^A$ (with $A$
fixed but arbitrary). We also show that the values of $s(n)$, for $n$
composite, are uniformly distributed among all $p$ residue classes modulo every
$p\le (\log{x})^A$. These appear to be the first results of their kind where
the modulus is allowed to grow substantially with $x$.
