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Mertens Constant

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

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

which is given by
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
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 $B_1$ also occurs in the Summatory Function of the number of Distinct Prime Factors,
\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

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 $\sigma_0(n)=\Omega(n)$,
\sum_{k=2}^n \Omega(k)=n\ln\ln n+B_2+o(n)
\end{displaymath} (6)

(Hardy and Wright 1979, p. 355).

See also Brun's Constant, Prime Number, Twin Primes Constant


Flajolet, P. and Vardi, I. ``Zeta Function Expansions of Classical Constants.'' Unpublished manuscript. 1996.

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.

© 1996-9 Eric W. Weisstein