= Factorial proof of Mertens first theorem
The <prime factorization> of $N!$ gives $\log N!=\sum_{p^j\le N}\lfloor N/p^j\rfloor\log p$. The <Chebyshev estimate> bounds the error on replacing each floor by its argument by $O(N)$. Dividing by $N$ and using the <Stirling formula> gives $\sum_{p^j\le N}(\log p)/p^j=\log N+O(1)$. The contribution of $j\ge2$ is bounded by the convergent series $\sum_{n\ge2}(\log n)/(n(n-1))$. This proves the <Mertens first theorem> without the <Prime number theorem>.
Back to article page
