Abstract
Abstract
We investigate the mean square behavior of error terms that arise in the summation of
gcd
(
n
1
,
…
,
n
k
)
$\gcd (n_1,\ldots ,n_k)$
gcd left parenthesis n 1 comma ellipsis comma n Subscript k Baseline right parenthesis
with
n
1
…
n
k
≤
x
$n_1\ldots n_k \le x$
n 1 ellipsis n Subscript k Baseline less than or equals x
. We first analyze error terms related to certain Dirichlet series
ζ
2
(
s
)
ζ
(
2
s
−
1
)
ζ
M
(
2
s
)
$\zeta ^2(s)\zeta (2s-1)\zeta ^M(2s)$
zeta squared left parenthesis s right parenthesis zeta left parenthesis 2 s minus 1 right parenthesis zeta Superscript upper M Baseline left parenthesis 2 s right parenthesis
and obtain upper bounds for their mean squares, depending on the parameter
M
. We then extend our analysis to the general case with
k
≥
3
$k \ge 3$
k greater than or equals 3
, providing estimates for the mean square of the corresponding error terms.