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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| Published in | Journal de theorie des nombres de bordeaux Vol. 21; no. 3; pp. 523 - 533 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Talence
cedram
01.01.2009
Université de Bordeaux 1, laboratoire de mathématiques pures |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| Author | DIAMOND, Harold G. PINTZ, Janos |
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| Copyright | Université Bordeaux 1, 2009 2015 INIST-CNRS |
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| Keywords | Analogy Sound Number theory Oscillation Inequality |
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| Snippet | 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... |
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| StartPage | 523 |
| SubjectTerms | 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 |
| Title | Oscillation of Mertens’ product formula |
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