2025.7.30 闲话

Let \(L(n)=\operatorname{lcm}(1,\cdots,n)\),

\[\ln\dfrac{L(n)}{L(n-1)}=\sum_{p\le n}\ln p\cdot(\lfloor\log_pn\rfloor-\lfloor\log_p(n-1)\rfloor)=\Lambda(n) \]

->

\[L(n)=\exp\sum_{i=1}^n\Lambda(i)=\exp\sum_{i=1}^n\mu(i)\sum_{k=1}^{\lfloor\frac ni\rfloor}\ln k=\prod_{i=1}^n\left\lfloor\dfrac ni\right\rfloor!^{\mu(i)} \]