  Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction
expansion of an irrational number $x\in (0, 1)$. 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 $S_n(x)=
\sum_{i=1}^n a_i(x)a_{i+1}(x)$ as well as, from a multifractal analysis point
of view, investigate its increasing rate. Specifically, we prove the following
results: \medskip \begin{itemize}
  \item For any $\epsilon>0$, the Lebesgue measure of the set $$\left\{x\in(0,
1): \left|\frac{ S_n(x)}{n\log^2 n}-\frac1{2\log2}\right|\geq
\epsilon\right\}$$tends to zero as $n$ to infinity.
  \item For Lebesgue almost all $x\in (0,1)$,
  $$\lim\limits_{n\rightarrow \infty} \frac{S_n(x)-\max\limits_{1\leq i \leq
n}a_i(x)a_{i+1}(x)}{n\log^2n}=\frac{1}{2\log2}.$$
  \item The Hausdorff dimension of the set
$$E(\phi):=\left\{x\in(0,1):\lim\limits_{n\rightarrow
\infty}\frac{S_n(x)}{\phi(n)}=1\right\}$$ is determined for a range of
increasing functions $\phi: \mathbb N\to \mathbb R^+$.
  \end{itemize}
