Oscillation of Mertens’ product formula
Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 523-533.

Mertens’ product formula asserts that

$\prod _{p\le x}\left(1-\frac{1}{p}\right)\phantom{\rule{0.166667em}{0ex}}logx\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{e}^{-\gamma }$

as $x\to \infty$. 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 $\pi \left(x\right)-\mathrm{li}x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.

La formule de Mertens affirme que

$\prod _{p\le x}\left(1-\frac{1}{p}\right)\phantom{\rule{0.166667em}{0ex}}logx\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}{e}^{-\gamma }$

quand $x\to \infty$. Les calculs montrent que la partie droite de la formule est supérieure à la partie gauche pour $2\le x\le {10}^{8}$. Par analogie avec le résultat de Littlewood sur $\pi \left(x\right)-\mathrm{li}x$, Rosser et Schoenfeld ont suggéré que cette inégalité et son contraire devait se produire pour des valeurs suffisamment grandes de $x$. Nous montrons que c’est bien le cas.

Published online:
DOI: 10.5802/jtnb.687
Classification: 11N37,  34K11
Keywords: Mertens’ product formula, oscillation, Euler’s constant, Riemann hypothesis, zeta function
Harold G. Diamond 1; Janos Pintz 2

1 Univ. of Illinois Dept. of Math. 1409 West Green Street Urbana, IL 61801 USA
2 Rényi Mathematical Institute Hungarian Academy of Sciences Reáltanoda u. 13-15 Budapest, H-1053, Hungary
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Harold G. Diamond; Janos Pintz. Oscillation of Mertens’ product formula. Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 523-533. doi : 10.5802/jtnb.687. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.687/

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