Oscillation of Mertens’ product formula

Diamond, Harold G.; Pintz, Janos

doi:10.5802/jtnb.687

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

Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 523-533.

Abstract
Résumé

Mertens’ product formula asserts that

\prod_{p \leq x} (1 - \frac{1}{p}) log x \to e^{- γ}

as $x \to \infty$ . Calculation shows that the right side of the formula exceeds the left side for $2 \leq x \leq 10^{8}$ . 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

\prod_{p \leq x} (1 - \frac{1}{p}) log x \to e^{- γ}

quand $x \to \infty$ . Les calculs montrent que la partie droite de la formule est supérieure à la partie gauche pour $2 \leq x \leq 10^{8}$ . 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.

MR Zbl

DOI: 10.5802/jtnb.687

Classification: 11N37, 34K11
Keywords: Mertens’ product formula, oscillation, Euler’s constant, Riemann hypothesis, zeta function

Author's affiliations:

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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     title = {Oscillation of {Mertens{\textquoteright}} product formula},
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     pages = {523--533},
     publisher = {Universit\'e Bordeaux 1},
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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://archive.numdam.org/articles/10.5802/jtnb.687/

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