bibkey: lichtman2020mertens authors: Jared Duker Lichtman year: 2020 title: Mertens’ prime product formula, dissected doi: null url: https://arxiv.org/html/2002.03361v3 claim: Theorem 1.1 recalls the reciprocal-prime sum with an O(1/log x) error and the classical Mertens product asymptotic; these inputs control prime tails at two linked cutoffs. strata_touched: [] license: citation-only triage: anchor

Mertens inputs for bounded arithmetic resolution

The inspected author version is arXiv:2002.03361v3. Theorem 1.1 is explicitly attributed to Mertens (1874). Its equation (1.1) includes

$$\sum _{p\le x}\frac{1}{p}=\log \log x+\beta +O(1/\log x),$$

and equation (1.2) states

$$P(x):=\prod _{p\le x}(1-1/p{)}^{-1}\sim e^{\gamma }\log x.$$

The symbol ${\log }_{2}x$ in that source means the iterated logarithm, not the base-two logarithm. The constant $\beta$ is the prime Mertens constant; it is distinct from the Euler–Mascheroni constant $\gamma$.

For linked cutoffs $m<X$, subtraction of (1.1) gives

$$\sum _{m<p\le X}\frac{1}{p}=\log \log X-\log \log m+O(1/\log m).$$

Replacing the summand by either $-\log (1-1/p)$ or $\log (1+1/p)$ costs $O({\sum }_{n>m}{n}^{-2})=O(1/m)$. Thus the reciprocal-prime estimate, including its error term, supplies the rate needed for the factorial-congruence comparison in FIB §§188–190. The product asymptotic alone suffices for convergence of $P(m(\log m{)}^2)/P(m)$ to one, but is not used to claim a sharper rate than its stated remainder provides.

The factorial-congruence comparison, its extremal witnesses and the FIB near-boundary neighborhood deductions are project paper arguments using these classical inputs. They are not assertions from Lichtman’s paper, not a priority claim, and not a formal proof of Robin or RH. No source text or proof implementation is copied into the library.