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2022 JHMT HS
10
GCD Double Infinite Sum
GCD Double Infinite Sum
Source:
August 8, 2024
number theory
greatest common divisor
2022
Problem Statement
Compute the exact value of
∑
a
=
1
∞
∑
b
=
1
∞
gcd
(
a
,
b
)
(
a
+
b
)
3
.
\sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{\gcd(a, b)}{(a + b)^3}.
a
=
1
∑
∞
b
=
1
∑
∞
(
a
+
b
)
3
g
cd
(
a
,
b
)
.
If necessary, you may express your answer in terms of the Riemann zeta function,
Z
(
s
)
=
∑
n
=
1
∞
1
n
s
Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
Z
(
s
)
=
∑
n
=
1
∞
n
s
1
, for integers
s
≥
2
s \geq 2
s
≥
2
.
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