Mertens Theorems

Mertens' theorems provide precise asymptotic formulas for sums and products over primes, refining our understanding of prime distribution.

TheoremMertens' First Theorem

\sum_{p \leq x} \frac{\log p}{p} = \log x + O(1)

This is equivalent to $\theta(x) \sim x$ and hence to the Prime Number Theorem.

TheoremMertens' Second Theorem

\sum_{p \leq x} \frac{1}{p} = \log \log x + M + O\left(\frac{1}{\log x}\right)

where $M \approx 0.2614972128...$ is the Meissel-Mertens constant.

TheoremMertens' Third Theorem

\prod_{p \leq x} \left(1 - \frac{1}{p}\right)^{-1} = e^{\gamma} \log x \left(1 + O\left(\frac{1}{\log x}\right)\right)

where $\gamma \approx 0.5772...$ is the Euler-Mascheroni constant.

ExampleInterpreting the Third Theorem

Since $\prod_p (1 - 1/p)^{-1}$ appears in the Euler product $\zeta(s) = \prod_p (1-p^{-s})^{-1}$ , Mertens' third theorem connects:

\prod_{p \leq x} \left(1 - \frac{1}{p}\right)^{-1} \sim e^{\gamma} \log x

to the singularity of $\zeta(s)$ at $s=1$ : $\zeta(s) \sim 1/(s-1)$ as $s \to 1^+$ .

Remark

These theorems can be proved using partial summation and the Prime Number Theorem. Conversely, Mertens' second theorem alone (without explicit error) is equivalent to PNT.

ExampleApplication to Probability

The probability that a random integer near $x$ is square-free is:

\prod_p \left(1 - \frac{1}{p^2}\right) = \frac{1}{\zeta(2)} = \frac{6}{\pi^2}

Mertens' theorem extends this, showing:

\prod_{p \leq x} \left(1 - \frac{1}{p}\right) \sim \frac{e^{-\gamma}}{\log x}

This is the "probability" that an integer near $x$ is coprime to all primes $\leq x$ , which by the sieve is approximately $1/\log x$ .

Mertens' theorems provide quantitative refinements of prime distribution, essential for applications in sieve theory and probabilistic number theory.