Oscillation of Mertens’ product formula
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 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.

DOI : https://doi.org/10.5802/jtnb.687
Classification:  11N37,  34K11
Keywords: Mertens’ product formula, oscillation, Euler’s constant, Riemann hypothesis, zeta function
@article{JTNB_2009__21_3_523_0,
author = {Diamond, Harold G. and Pintz, Janos},
title = {Oscillation of Mertens' product formula},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux 1},
volume = {21},
number = {3},
year = {2009},
pages = {523-533},
doi = {10.5802/jtnb.687},
mrnumber = {2605532},
zbl = {1214.11102},
language = {en},
url = {http://www.numdam.org/item/JTNB_2009__21_3_523_0}
}

Diamond, Harold G.; Pintz, Janos. 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. http://www.numdam.org/item/JTNB_2009__21_3_523_0/

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