Landau Symbols

Let  be an integer variable which tends to infinity and let  be a continuous variable tending to some limit. Also, let  or  be a positive function and  or  any function. Then the symbols  (sometimes called "big-O") and  (sometimes called "little-o") are known as the Landau symbols and defined as follows.

1.  means that  for some constant  and all values of  and ,

2.  means that 

(Hardy and Wright 1979, pp. 7-8).

Historically speaking, the symbol  first appeared in the second volume of Bachmann's treatise on number theory (Bachmann 1894), and Landau obtained this notation in Bachmann's book (Landau 1909, p. 883; Derbyshire 2004, p. 238). However, the symbol  did indeed originate with Landau (1909) in place of the earlier notation (Narkiewicz 2000, p. XI).

REFERENCES:

Bachmann, P. Analytische Zahlentheorie, Bd. 2: Die Analytische Zahlentheorie. Leipzig, Germany: Teubner, 1894.

de Bruijn, N. G. Asymptotic Methods in Analysis. New York: Dover, pp. 3-10, 1981.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Hardy, G. H. and Wright, E. M. "Some Notations." §1.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 7-8, 1979.

Havil, J. "Big Oh Notation." Appendix B in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 219, 2003.

Miller, J. "Earliest Uses of Symbols of Number Theory." http://members.aol.com/jeff570/nth.html.

Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig, Germany: Teubner, 1909. Reprinted by New York: Chelsea, 1953.

Narkiewicz, W. The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. New York: Springer-Verlag, 2000.