Mertens — Compactness of $M(x)$

Statement

Mertens’ theorem (1874) states that the sum of reciprocals of primes satisfies

where $B \approx 0.26149\ldots$ is Mertens’ constant. In particular, the function $M(x) - \log\log x$ remains uniformly bounded for $x \geq 2$.

Why it matters (role in PT)

Mertens is a classical theorem (analytic number theory, independent of PT). PT imports it for a precise purpose: compactness of the sieve transfer matrices.

At each sieve level $k$, the auto-transition rate $\alpha_k$ satisfies $\alpha_k \to 1/2$ by Mertens (equivalently by the prime number theorem). The stochastic matrices $T(k)$ therefore remain in a compact set, and any continuous functional of $T(k)$ converges.

This is exactly what the proof of T4 (spectral convergence) requires: without Mertens, one could not conclude that eigenvectors and the spectral radius converge.

No circularity. Mertens provides the background asymptotic $\alpha_k = 1/2 - O(1/\ln p_k)$ — valid for any multiplicative sieve. T4 proves the PT-specific structural content (spectral annihilation $r_2(0) = 0$, CRT dilution); the two bricks are logically independent.

Proof — outline

Classical (elementary) proof. Partial summation against Chebyshev’s estimate $\sum_{p \leq x} \log p / p = \log x + O(1)$ yields $M(x) = \log\log x + B + O(1/\log x)$. See Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, ch. I.1.

See also

• T4 — Spectral convergence — main user of Mertens
• T5 — Attractor $\mu^* = 15$ — uses T4
• Active prime criterion — discriminates $\gamma_p \gtrless s$
• All theorems