We prove that for any FAb compact p-adic analytic group
G, its representation zeta function is a finite sum of terms ni−s fi(p−s), where ni are natural numbers and fi(t) ∈ Q(t) are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If G is moreover a pro-p group, we prove that its representation zeta function is rational in p−s. These results were proved by Jaikin-Zapirain for p > 2 or for G uniform and pro-2,
respectively. We give a new proof which avoids the Kirillov orbit method and works for all p.