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Meissel–Mertens constant
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(Redirected from Mertens constant)
n |
-
B1 := k∑ i = 1
− log log pk =1 pi π (n)∑ i = 1
− log log n,1 pi
and
-
B1 = γ + k∑ i = 1
− log1 pi k∏ i = 1
=1 1 − 1 pi γ + k∑ i = 1
− log1 pi
= γ +1 1 − 1 pi ∑ p
p prime
+ log 1 −1 p
,1 p
π (n) |
pi |
i |
γ |
Mertens’ second theorem asserts that the limit exists. (Is it known whether the Meissel–Mertens constant is rational or irrational?)
The fact that there are two logarithms ( log log) in the limit for the Meissel–Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler–Mascheroni constant.
Contents
Decimal expansion of the Meissel–Mertens constant
The decimal expansion of the Meissel–Mertens constantB1 |
-
B1 = 0.2614972128476427837554268386086958590516....
B1 := lim k → ∞ (
|
pi |
i |
- {2, 6, 1, 4, 9, 7, 2, 1, 2, 8, 4, 7, 6, 4, 2, 7, 8, 3, 7, 5, 5, 4, 2, 6, 8, 3, 8, 6, 0, 8, 6, 9, 5, 8, 5, 9, 0, 5, 1, 5, 6, 6, 6, 4, 8, 2, 6, 1, 1, 9, 9, 2, 0, 6, 1, 9, 2, 0, 6, 4, 2, 1, 3, 9, 2, 4, 9, ...}
Continued fraction for the Meissel–Mertens constant
The simple continued fraction for the Meissel–Mertens constant is
B1 = 0 +
|
A230767 Continued fraction for the Meissel–Mertens constant.
- {0, 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, ...}
Noncomposite reciprocal constant
The noncomposite reciprocal constant is given by
-
C := k∑ i = 1
− log log qk =1 qi ∑ q ≤ n
q noncomposite
− log log n = γ − C = B1 + 1,1 q
qi |
i |
C |
Decimal expansion of the noncomposite reciprocal constant
The decimal expansion of the noncomposite reciprocal constant is
-
C = γ − C = B1 + 1 = 1.2614972128476427837554268386086958590516...
See also
External links
- Weisstein, Eric W., Mertens Constant, from MathWorld—A Wolfram Web Resource.
- Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens (pdf).
- Divergence of the sum of the reciprocals of the primes—Wikipedia.org.