Mertens Constant

A constant related to the Twin Primes Constant which appears in the Formula for the sum of inverse Primes

$\begin{displaymath} \sum_{p{\rm\ prime}}^x {1\over p}=\ln\ln x+B_1+o(1) \end{displaymath}$

(1)

which is given by

$\begin{displaymath} B_1=\gamma+\sum_{p{\rm\ prime}} \left[{\ln(1-p^{-1})+{1\over p}}\right]\approx 0.261497. \end{displaymath}$

(2)

Flajolet and Vardi (1996) show that

$\begin{displaymath} e^{B_1}=e^\gamma\prod_{m=2}^\infty \zeta(m)^{\mu(m)/m}, \end{displaymath}$

(3)

where $\gamma$ is the Euler-Mascheroni Constant, $\zeta(n)$ is the Riemann Zeta Function, and $\mu(n)$ is the Möbius Function. The constant

also occurs in the Summatory Function of the number of Distinct Prime Factors,

$\begin{displaymath} \sum_{k=2}^n \omega(k)=n\ln\ln n+B_1 n+o(n) \end{displaymath}$

(4)

(Hardy and Wright 1979, p. 355).

The related constant

$\begin{displaymath} B_2=\gamma+\sum_{p{\rm\ prime}} \left[{\ln(1-p^{-1})+{1\over p-1}}\right]\approx 1.034653 \end{displaymath}$

(5)

appears in the Summatory Function of the Divisor Function $\sigma_0(n)=\Omega(n)$ ,

$\begin{displaymath} \sum_{k=2}^n \Omega(k)=n\ln\ln n+B_2+o(n) \end{displaymath}$

(6)

(Hardy and Wright 1979, p. 355).

References

Flajolet, P. and Vardi, I. ``Zeta Function Expansions of Classical Constants.'' Unpublished manuscript. 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps.

Hardy, G. H. and Weight, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 351 and 355, 1979.