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Using estimates on Hooley’s [Formula: see text]-function and a short interval version of the celebrated Dirichlet hyperbola principle, we derive an asymptotic formula for a class of arithmetic functions over short segments. Numerous examples are also given.

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions., Comment: 8 pages

We obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} \varphi \left( \left\lfloor{x/n}\right\rfloor\right)$, involving the Euler functions $\varphi$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the reciprocals of integers., Comment: This version incorporate the improvement of the upper bound of Theorem 2.1 in v.1, given by Lixia Dai and Hao Pan in arXiv:1809.10381. Consecutively, Lixia Dai and Hao Pan are now co-authors

Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl's bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials., 6 pages