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

px1-1plogxe-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.

La formule de Mertens affirme que

px1-1plogxe-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.

Received:
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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