P5221

欧拉函数

\[\prod_{i = 1}^N\prod_{j = 1}^N\frac{\text{lcm}(i, j)}{\gcd(i, j)}\\ \]

\[=\prod_{i = 1}^N\prod_{j = 1}^N\frac{i\times j}{\gcd(i, j)^2}\\ \]

\[=(\prod_{i = 1}^N\prod_{j = 1}^Ni\times j)\times(\prod_{i = 1}^N\prod_{j = 1}^N\gcd(i, j))^{-2}\\ \]

\[\prod_{i = 1}^N\prod_{j = 1}^Ni\times j = (n!)^{2n}\\ \]

\[\prod_{i = 1}^N\prod_{j = 1}^N\gcd(i, j)\\ \]

\[=\prod_{d = 1}^N\prod_{i = 1}^N\prod_{j = 1}^N[\gcd(i, j) = d]\\ \]

\[=\prod_{d = 1}^Nd^{\sum_{i = 1}^N\sum_{j = 1}^N[\gcd(i, j) = d]}\\ \]

\[=\prod_{d = 1}^Nd^{\sum_{i = 1}^{\frac{N}{d}}\sum_{j = 1}^{\frac{N}{d}}[\gcd(i, j) = 1]}\\ \]

\[=\prod_{d = 1}^Nd^{2\times \sum_{i = 1}^{\left \lfloor \frac{N}{d}\right \rfloor} \varphi(i) - 1}\\ \]

\[s_i = \sum_{i = 1}^i \varphi(i)\\ \]

\[\prod_{i = 1}^N\prod_{j = 1}^N\gcd(i, j)\\ \]

\[=\prod_{d = 1}^Nd^{2\times s_{\left \lfloor \frac{N}{d}\right \rfloor} - 1}\\ \]

\[=\prod_{i = 1}^N\prod_{j = 1}^N\frac{\text{lcm}(i, j)}{\gcd(i, j)}\\ \]

\[=(n!)^{2n}\times (\prod_{d = 1}^Nd^{2\times s_{\left \lfloor \frac{N}{d}\right \rfloor} - 1})^{-2} \]