Oscillation of Mertens’ product formula

Mertens’ product formula asserts that ∏ p ≤ x ( 1 − 1 p ) log   x → e − γ asx→ ∞. Calculation shows that the right side of the formula exceeds the left side for 2 ≤x≤ 10⁸. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result onπ(x)– lix, this and a complementary inequa...

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Bibliographic Details
Published inJournal de theorie des nombres de bordeaux Vol. 21; no. 3; pp. 523 - 533
Main Authors DIAMOND, Harold G., PINTZ, Janos
Format Journal Article
LanguageEnglish
Published Talence cedram 01.01.2009
Université de Bordeaux 1, laboratoire de mathématiques pures
Subjects
Algebra
Exact sciences and technology
Infinity
Integration by parts
Log integral function
Logical givens
Mathematical theorems
Mathematics
Mellin transforms
Number theory
Prime numbers
Sciences and techniques of general use
Zero
Analogy
Sound
Number theory
Oscillation
Inequality
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Summary:Mertens’ product formula asserts that ∏ p ≤ x ( 1 − 1 p ) log   x → e − γ asx→ ∞. Calculation shows that the right side of the formula exceeds the left side for 2 ≤x≤ 10⁸. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result onπ(x)– lix, this and a complementary inequality might change their sense for sufficiently large values ofx. We show this to be the case.
ISSN:1246-7405
2118-8572
DOI:10.5802/jtnb.687