Mertens' theorems

Remarks

For the first theorem, compare two expressions of $\log(n!)$. Legendre's formula gives $\log(n!) = \sum_{p^k \le n} \floor{n/p^k} \log p = n \sum_{p \le n} \frac{\log p}{p} + O(n)$, using the bound $\psi(n) = O(n)$ of Chebyshev for the Chebyshev function $\psi$. Stirling's formula gives $\log(n!) = n \log n + O(n)$, and dividing by $n$ gives the result. The second theorem follows from the first by partial summation. The third follows from the second by taking logarithms; identifying the constant $e^{-\gamma}$ needs the behaviour of the Riemann zeta function near $s = 1$. The constant $M \approx 0.2615$ is the Meissel–Mertens constant. The theorems do not need the prime number theorem; they are elementary.

Going deeper

For a student

The theorems are typical of elementary prime number theory: they use only Chebyshev bounds and the factorization of $n!$. The third theorem is used in sieve methods, where products over primes such as $\prod_{p \le z} (1 - 1/p)$ give the main term. A naive sieve heuristic with $z = \sqrt{x}$ would predict about $2e^{-\gamma} x/\log x \approx 1.12\, x/\log x$ primes up to $x$, while the prime number theorem gives $x/\log x$. The difference shows that divisibility by different large primes is not independent.