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Abstract. This paper presents asymptotic formulas for the abundance of binomial, generalized binomial, multinomial, and generalized multinomial coefficients having any given degree of prime-power divisibility. In the case of binomial coefficients, for a fixed prime p, we consider the number of (x, y) with 0 ≤ x, y < p n for which x+y x is divisible by p zn (but not p zn+1 ) when zn is an integer and α < z < β, say. By means of a classical theorem of Kummer and the probabilistic theory of large deviations, we show that this number is approximately p nD((α,β)) , where D((α, β)) := sup{D(z) : α < z < β} and D is given by an explicit formula. We also develop a "p-adic multifractal" theory and show how D may be interpreted as a multifractal spectrum of divisibility dimensions. We then prove that essentially the same results hold for a large class of the generalized binomial coefficients of Knuth and Wilf, including the q-binomial coefficients of Gauss and the Fibonomial coefficients of Lucas, and finally we extend our results to multinomial coefficients and generalized multinomial coefficients.

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Let Z(t) be the population size at time t in a general age-dependent branching process (as defined by Crump and Mode, or Jagers) in which the number N of offspring of a parent has expected value 1 (critical case). Assuming positivity and finiteness of the second moments of N, of the lifespan distribution and of the expected number of births per parent as a function of age (also assumed to be strongly non-lattice), the distribution of Z(t)/t conditioned on non-extinction at time t is asymptotically exponential. The main step in the proof is a comparison lemma for the probability generating functions of Z(t) and of the embedded generation process.