CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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A Theta(1/log x) product bound does not determine the Mertens constant

Statement refuted

Knowing only that a positive function $F (x)$ satisfies

F (x) = Θ (1 / \log x)

determines the exact leading constant in front of $1 / \log x$ .

Facts & Assumptions

Given: The weaker $Θ (1 / \log x)$ conclusion of Shoup's product bound and the exact constant statement of Mertens' third theorem for primes.

[L1]

The second and third Mertens theorems distinguish a bounded-error reciprocal-prime asymptotic from the exact factor $e^{- γ}$ in the product formula (Mertens' second theorem for primes, Mertens' third theorem for primes).

Counterexample

technique · direct

1.1

The two positive functions $F_{1} (x) : = \frac{1}{\log x}, F_{2} (x) : = \frac{2}{\log x}$ both satisfy $F_{j} (x) = Θ (1 / \log x)$ as $x \to \infty$ .

givenconstruct

2.1

Their leading constants are different: one is $1$ and the other is $2$ . So a mere $Θ (1 / \log x)$ estimate leaves the multiplicative constant free. What Mertens' third theorem for primes adds over that weaker statement is exactly the identification of the constant as $e^{- γ}$ .

L1step 1.1algebra∎

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Dependency tree · two levels

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Sources

Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 2 (standard reference, not scraped)
Terence Tao, Mertens' theorems (standard reference, not scraped)