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# Meissel–Mertens constant

(Redirected from Harmonic series of the primes)

The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker’s constant, Hadamard–de la Vallée-Poussin constant, or prime reciprocal constant, is a mathematical constant in number theory, defined as the limiting difference between the harmonic series of the primes and the natural logarithm of the natural logarithm of
 n
, i.e.
B1 :=
 k ∑ i   = 1

 1 pi

− log  log pk
=
 π (n) ∑ i   = 1

 1 pi

− log  log n
,

and

B1  =  γ +
 k ∑ i   = 1

 1 pi

− log
 k ∏ i   = 1

1
1 −
 1 pi

=
γ +
 k ∑ i   = 1

 1 pi
− log
1
1 −
 1 pi

=  γ +
 ∑ pp prime

 1 p
+ log 1 −
 1 p

,
where
 π (n)
is the prime counting function,
 pi
is the
 i
-th prime and
 γ
is the Euler–Mascheroni constant, which is the analogous constant involving the harmonic series (which is over all positive integers).

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.

## Decimal expansion of the Meissel–Mertens constant

The decimal expansion of the Meissel–Mertens constant
 B1
is
 B1 = 0.2614972128476427837554268386086958590516....
A077761 Decimal expansion of Mertens’ constant,
B1   :=   limk →   (
 k

 i   = 1
 1 pi
−  log  log pk  )
, where
 pi
is the
 i
-th prime number.
{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 +
1
3 +
1
1 +
1
4 +
1
1 +
1
2 +
1
5 +
 1 ⋱

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

 1 qi

− log  log qk
=
 ∑ q  ≤  nq noncomposite

 1 q

− log  log n
=  γC  =  B1 + 1,
where
 qi
is the
 i
-th [positive] noncomposite (one or prime) and
 C
is the composite reciprocal constant.

### Decimal expansion of the noncomposite reciprocal constant

The decimal expansion of the noncomposite reciprocal constant is

 C  =  γ − C  =  B1 + 1  =  1.2614972128476427837554268386086958590516...