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

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.

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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