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Harold G. Diamond;
Janos PintzOscillation of Mertens’ product formulaJournal de théorie des nombres de Bordeaux,
21 no.
3 (
2009), p. 523-533, doi:
10.5802/jtnb.687
Article
PDF | Reviews
MR 2605532 |
Zbl 1214.11102
Class. Math.:
11N37,
34K11
Keywords: Mertens’ product formula, oscillation, Euler’s constant, Riemann hypothesis, zeta function
Mertens’ product formula asserts that
$$ \prod _{p \le x} \Big ( 1 - \frac{1}{p} \Big )\, \log x \, \rightarrow \, e^{-\gamma } $$
as $x \rightarrow \infty $. Calculation shows that the right side of the formula exceeds the left side for $2 \le x \le 10^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi (x) - \textrm{li } x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.
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