Hadamard-Vallée Poussin Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

The sum of Reciprocals of Primes diverges, but

$$\lim_{n\to\infty}\left[{\sum_{k=1}^{\pi(n)} {1\over p_k}-\ln(\ln n)}\right] = \gamma+\sum_{k=1}^\infty \left[{\ln\left({1-{1\over p_k}}\right)+{1\over p_k}}\right]$$

$$\equiv C_1=0.2614972128\ldots,$$ (1)

where $\pi(n)$ is the Prime Counting Function and $\gamma$ is the Euler-Mascheroni Constant (Le Lionnais 1983). Hardy and Wright (1985) show that, if $\omega(n)$ is the number of distinct Prime factors of $n$, then

$$\lim_{n\to\infty}\left[{{1\over n}\sum_{k=1}^n \omega(k)-\ln(\ln n)}\right]=C_1.$$ (2)

Furthermore, if $\Omega(n)$ is the total number of Prime factors of $n$, then

$$\lim_{n\to\infty}\left[{{1\over n}\sum_{k=1}^n \Omega(k)-\ln... ..._1+\sum_{k=1}^\infty {1\over p_k(p_k-1)} = 1.0346538819\ldots.$$ (3)

Similarly,

$$\lim_{n\to\infty}\left({\,\sum_{k=1}^{\pi(n)} {\ln p_k\over p_k}-\ln n}\right) = -\gamma-\sum_{j=2}^\infty \sum_{k=1}^\infty {\ln p_k\over {p_k}^j}$$

$$\equiv -C_2 = -1.3325822757\ldots.$$ (4)

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/hdmrd/hdmrd.html

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1985.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 24, 1983.

Rosser, J. B. and Schoenfeld, L. ``Approximate Formulas for Some Functions of Prime Numbers.'' Ill. J. Math. 6, 64-94, 1962.