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

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.

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.

Reçu le : 2008-08-30
Publié le : 2010-03-22
DOI : https://doi.org/10.5802/jtnb.687
Classification : 11N37,  34K11
Mots clés : Mertens’ product formula, oscillation, Euler’s constant, Riemann hypothesis, zeta function
@article{JTNB_2009__21_3_523_0,
     author = {Harold G. Diamond and Janos Pintz},
     title = {Oscillation of Mertens' product formula},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {523--533},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.687},
     zbl = {1214.11102},
     mrnumber = {2605532},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2009__21_3_523_0/}
}
Harold G. Diamond; Janos Pintz. 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. https://jtnb.centre-mersenne.org/item/JTNB_2009__21_3_523_0/

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