Metric results on summatory arithmetic functions on Beatty sets

Cancellation of two classes of dirichlet coefficients over Beatty sequences

“…Moreover, by (1.2), the result of Technau and Zafeiropoulos can improve error terms in Theorem 1.1, Corollary 1.3 and Theorem 1.6. However, the results of [1] and [34] yield no further information on the set of admissible α > 1 other than having full Lebesgue measure and are, therefore, rather less explicit than our results which apply for all α of finite type (see Section 2).…”

“…For all α > 1 in a set of full Lebesgue measure (depending on f ), Technau and Zafeiropoulos [34] have proved…”

On the Divisor Function Over Nonhomogeneous Beatty Sequences

“…where the implied constant may depend on α and ε. This result was subsequently improved and extended in various ways (see [2,9,11,14]). Zhai [14] proved that for almost all α > 1 with respect to the Lebesgue measure,…”

“…There are precise estimates for sums of the divisor function over homogeneous Beatty sequences. Abercrombie proved in [1] that for almost all with respect to the Lebesgue measure, where the implied constant may depend on and This result was subsequently improved and extended in various ways (see [2, 9, 11, 14]). Zhai [14] proved that for almost all with respect to the Lebesgue measure, where the implied constant may depend on and In fact, this result can be modified to apply to an individual …”

On $k$-free numbers over Beatty sequences

“…Recently, with some much more generalized arithmetic functions, in [1,11], one may also get some other estimates for such type sums. However, the estimates of [1,11] cannot be applied to an individual α. In this paper, we can give the following formula.…”