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2011 Princeton University Math Competition
A5 / B8
A5 / B8
Part of
2011 Princeton University Math Competition
Problems
(1)
2011 PUMaC Number Theory A5 / B8
Source:
9/24/2019
Let
d
(
n
)
d(n)
d
(
n
)
denote the number of divisors of
n
n
n
(including itself). You are given that
∑
n
=
1
∞
1
n
2
=
π
2
6
.
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.
n
=
1
∑
∞
n
2
1
=
6
π
2
.
Find
p
(
6
)
p(6)
p
(
6
)
, where
p
(
x
)
p(x)
p
(
x
)
is the unique polynomial with rational coefficients satisfying
p
(
π
)
=
∑
n
=
1
∞
d
(
n
)
n
2
.
p(\pi) = \sum_{n=1}^{\infty} \frac{d(n)}{n^2}.
p
(
π
)
=
n
=
1
∑
∞
n
2
d
(
n
)
.
number theory