On the fractional [parts connecter with the function $N/x$

The Dirichlet divisor problem is closely related to the sum of fractional parts
$$\sum_{n\le N}\left\{\frac{x}{n}\right\}\,=\,cN\,+\,O\bigl(x^{\alpha+\varepsilon}\bigr).$$
In the case $N \le x^\beta$, $0.5\le \beta < 1$, the fractional parts are uniformly distributed over $[0, 1)$ and therefore $c=\tfrac{1}{2}$. However, in the case $N = x$ the distribution is not uniform, since $c=1-\gamma=0.422784\ldots$ ($\gamma$ is Euler constant).

In the talk, we will consider some asymptotic formulas for general sums
$$\sum_{\substack{n \le x \\ n \in \mathcal{A}}} f\left(\left\{\frac{x}{n}\right\}\right), \quad \sum_{n \le x}g(n)f\left(\left\{\frac{x}{n}\right\}\right),$$
where $\mathcal{A}$ denotes some subset of natural numbers, and $f$, $g$ are real-valued functions satisfying some natural conditions. We also give some applications of these formulas.