Affiliation:
1. Department of Mathematics Faculty of Science Kasetsart University Bangkok , Bangkok 10900 Thailand
Abstract
Abstract
Let
f
k
(
n
)
:=
∑
d
k
∣
n
Φ
(
d
)
,
$$
f_k(n):=\sum\limits_{d^k \mid n} \Phi(d),
$$
where Φ(d) is a multiplicative function, Φ(d) = O(dε
). We study asymptotic behaviour of the sum
T
f
k
c
(
N
)
:=
∑
n
≤
N
f
k
n
c
,
1
<
c
<
2
,
$$
T_{f_k}^c(N):=\sum\limits_{n \leq N} f_k\left(\left\lfloor n^c\right\rfloor\right), \quad 1<c<2,
$$
and establish some asymptotic formulas for
T
f
k
c
(
N
)
$
T_{f_k}^c(N)
$
under assumptions on fk
.
Reference11 articles.
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2. CAO, X. — ZHAI, W.: The distribution of square-free numbers of the form [nc], J. Théor. Nombres Bordeaux 10.2 (1998), 287–299.
3. CAO, X. — ZHAI, W.: On the distribution of square-free numbers of the form bncc (II), Acta Math. Sin. (Chin. Ser) 51 (2008), 1187–1194 (in Chinese).
4. DESHOUILLERS, J. M.: A remark on cube-free numbers in Segal–Piatestki-Shapiro sequences, Hardy-Ramanujan J. 41 (2019), 127–132.
5. PYATETSKII-SHAPIRO, I. I.: On the distribution of prime numbers in sequences of the form fn), Mat. Sb. 33 (1953), 559–566.
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1 articles.
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