AbstractLet $$ \gcd (k,j) $$
gcd
(
k
,
j
)
denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define $$\begin{aligned} M_r(x; f) := \sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^{k}j^{r}f(\gcd (j,k)) \end{aligned}$$
M
r
(
x
;
f
)
:
=
∑
k
≤
x
1
k
r
+
1
∑
j
=
1
k
j
r
f
(
gcd
(
j
,
k
)
)
for any large real number $$x\ge 5$$
x
≥
5
, where f is any arithmetical function. Let $$\phi $$
ϕ
, and $$\psi $$
ψ
denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $$M_r(x; \mathrm{id})$$
M
r
(
x
;
id
)
, $$M_r(x;{\phi })$$
M
r
(
x
;
ϕ
)
and $$M_r(x;{\psi })$$
M
r
(
x
;
ψ
)
. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $$M_r(x;\mathrm{id})$$
M
r
(
x
;
id
)
for any large positive number $$x>5$$
x
>
5
satisfying $$x=[x]+\frac{1}{2}$$
x
=
[
x
]
+
1
2
.
On Vandiver’s arithmetical function – I
