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

 (1)

where is the Prime Counting Function and is the Euler-Mascheroni Constant (Le Lionnais 1983). Hardy and Wright (1985) show that, if is the number of distinct Prime factors of , then
 (2)

Furthermore, if is the total number of Prime factors of , then

 (3)

Similarly,

 (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.