From the introduction: Let \(\{X_i\}_{i\geq 1}\) be a sequence of independent random variables taking the values \(\pm 1\) with the probability \({1 \over 2}\), and let us set \(S_n=X_1+X_2+\cdots+X_n\). A classical theorem of Hardy and Littlewood (1914) says that, for any \(C>0\) and for all \(n\) large enough, we have \[ S_n\leq C\sqrt {n\log n},\tag{1} \] with probability 1. In 1924, Khinchin showed that (1) can be replaced by a sharper inequality \[ S_n\leq\sqrt {(2+ \varepsilon) n\log\log n},\tag{2} \] for any \(\varepsilon>0\). In view of Khinchin’s result, inequality (1) has long been considered as one of a rather historical value. However, the recent results on Brownian motion on Riemannian manifolds give a new insight into it. In this paper, we show that an analogue of (1), for the Brownian motion on Riemannian manifolds of the polynomial volume growth, is sharp and, therefore, cannot be replaced by an analogue of (2).