Given an integer $n$, the Jacobsthal function $g(n)$ is the least integer, so that among any $g(n)$ consecutive integers $a,a+1,\dots,a+g(n)-1$ there is at least one that is coprime to $n$. Let $\nu(n)$ count the distinct prime factors of $n$. You can define $$C(r)=\max_{\nu(n)=r} g(n)$$ and as Jonas Meyer points out in the comments this is precisely $C(t)=\gamma (t)$ (i.e. it is enough to consider when all $a_i$ are prime).
For the bounds $$\frac{c_1r \frac{c_1t (\log r)^2 t)^2 \log \log \log r}{(\log\log r)^2}\le t}{(\log\log t)^2}\le C(t)\le c_2 t^{c_3}$$ see the paper "On the integers relatively prime to n and on a number-theoretic function considered by Jacobsthal"" by Erdos. I don't know if there are better bounds.
Given an integer $n$, the Jacobsthal function $g(n)$ is the least integer, so that among any $g(n)$ consecutive integers $a,a+1,\dots,a+g(n)-1$ there is at least one that is coprime to $n$. Let $\nu(n)$ count the distinct prime factors of $n$. You can define $$C(r)=\max_{\nu(n)=r} g(n)$$ and as Jonas Meyer points out in the comments this is precisely $C(t)=\gamma (t)$ (i.e. it is enough to consider when all $a_i$ are prime).
For the bounds $$\frac{c_1r (\log r)^2 \log \log \log r}{(\log\log r)^2}\le C(t)\le c_2 t^{c_3}$$ see the paper "On the integers relatively prime to n and on a number-theoretic function considered by Jacobsthal"" by Erdos. I don't know if there are better bounds.