Oscillation of Mertens’ product formula
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 523-533.

La formule de Mertens affirme que

px 1 - 1 plogxe -γ

quand x. Les calculs montrent que la partie droite de la formule est supérieure à la partie gauche pour 2x10 8 . Par analogie avec le résultat de Littlewood sur π(x)-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.

Mertens’ product formula asserts that

px 1 - 1 plogxe -γ

as x. Calculation shows that the right side of the formula exceeds the left side for 2x10 8 . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on π(x)-li x, this and a complementary inequality might change their sense for sufficiently large values of x. We show this to be the case.

DOI : 10.5802/jtnb.687
Classification : 11N37, 34K11
Mots clés : Mertens’ product formula, oscillation, Euler’s constant, Riemann hypothesis, zeta function
Diamond, Harold G. 1 ; Pintz, Janos 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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Diamond, Harold G.; Pintz, Janos. Oscillation of Mertens’ product formula. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 523-533. doi : 10.5802/jtnb.687. http://archive.numdam.org/articles/10.5802/jtnb.687/

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