Let [a 1(x), a 2(x), …, a n (x), …] be the continued fraction expansion of an irrational number x ∈ (0, 1). The study of the growth rate of the product of consecutive partial quotients a n (x)a n+1(x) is associated with the improvements to Dirichlet’s theorem (1842). We establish both the weak and strong laws of large numbers for the partial sums as well as, from a multifractal analysis point of view, investigate its increasing rate. Specifically, we prove the following results:• For any ϵ > 0, the Lebesgue measure of the set tends to zero as n to infinity.• For Lebesgue almost all x ∈ (0, 1), • The Hausdorff dimension of the set is determined for a range of increasing functions .