Oscillation of Mertens’ product formula. (English) Zbl 1214.11102

In 1874, Franz Mertens [J. Reine Angew. Math. 78, 46–62 (1874; JFM 06.0116.01)] proved the following asymptotic formula \[ \lim_{x \to \infty} (\log x) \prod_{p\leq x} \left( 1 - {1 \over p}\right) = e^{-\gamma} \] where the product is over primes \(p \leq x\) and \(\gamma\) is Euler’s constant. In 1962, calculations of J. B. Rosser and L. Schoenfeld [Ill. J. Math. 6, 64–94 (1962; Zbl 0122.05001)] revealed that \[ {e^{-\gamma} \over \log x + 2/\sqrt{x}} < \prod_{p\leq x} \left( 1 - {1 \over p}\right) < {e^{-\gamma} \over \log x}, \] for \(2\leq x < 10^8\). In analogy with Littlewood’s famous result about sign changes of \(\pi(x) - {\text{li } }x\), they asked if the above inequalities may also change their sign infinitely often. The paper under review shows that this is the case. More precisely, the authors show that the quantity \[ \sqrt{x} \left( \prod_{p\leq x} \left(1 - \frac{1}{p}\right)^{-1} - e^\gamma \log x \right) \] attains large positive and negative values as \(x\) tends to infinity. The key ingredient in the proof is Ingham’s version of the Wiener-Ikehara Tauberian theorem.


11N05 Distribution of primes
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M45 Tauberian theorems
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