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 the prime reciprocal constant , is a mathematical constant in number theory , defined as the limiting difference between the harmonic series summed only over the primes and the natural logarithm of the natural logarithm:
M
=
lim
n
→
∞
(
∑
p
prime
p
≤
n
1
p
−
ln
(
ln
n
)
)
=
γ
+
∑
p
[
ln
(
1
−
1
p
)
+
1
p
]
.
{\displaystyle M=\lim _{n\rightarrow \infty }\left(\sum _{\scriptstyle p{\text{ prime}} \atop \scriptstyle p\leq n}{\frac {1}{p}}-\ln(\ln n)\right)=\gamma +\sum _{p}\left[\ln \!\left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right].}
In the limit, the sum of the reciprocals of the primes < n and the function ln(ln n ) are separated by a constant, the Meissel–Mertens constant (labelled M above).
Here γ is the Euler–Mascheroni constant , which has an analogous definition involving a sum over all integers (not just the primes).
The plot of the prime harmonic sum up to
n
=
2
15
,
2
16
,
…
,
2
46
≈
7.04
×
10
13
{\displaystyle n=2^{15},2^{16},\ldots ,2^{46}\approx 7.04\times 10^{13}}
and the Merten's approximation to it. The original of this figure has y axis of the length 8 cm and spans the interval (2.5, 3.8), so if the n axis would be plotted in the linear scale instead of logarithmic, then it should be
5.33
(
3
)
×
10
9
{\displaystyle 5.33(3)\times 10^{9}}
km long — that is the size of the Solar System.
The value of M is approximately
M ≈ 0.2614972128476427837554268386086958590516... (sequence A077761 in the OEIS ) .
Mertens' second theorem establishes that the limit exists.
The fact that there are two logarithms (log of a 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.