  The Shintani cocycle on $\GL_n(\Q)$, as constructed by Hill, gives a
cohomological interpretation of special values of zeta functions for totally
real fields of degree $n$. We give an explicit criterion for a specialization
of the Shintani cocycle to be $p$-adically interpolable. As a corollary, we
recover the results of Deligne-Ribet, Cassou Nogu\`es and Barsky on the
construction of $p$-adic $L$-functions attached to totally real fields.
