  We conjecture average counting functions for prime $k$-tuples based on a
gamma distribution hypothesis for prime powers. The conjecture is closely
related to the Hardy-Littlewood conjecture for $k$-tuples but yields better
estimates. Possessing average counting functions along with their corresponding
exact counting functions allows to implicitly define pertinent $k$-tuple zeta
functions. The $k$-tuple zeta functions in turn allow construction of $k$-tuple
analogs of explicit formulae. If the zeros of the (implicitly defined)
$k$-tuple zeta can be determined, the explicit formulae should yield a
(dis)proof of the $k$-tuple analog of the prime number theorem.
