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Meissel–Mertens constant
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(Redirected from Harmonic series of the primes)
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.