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# 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).

Asymptotic Notation, Big-Omega Notation, Big-Theta Notation

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## 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.

Landau Symbols

## Cite this as:

Weisstein, Eric W. "Landau Symbols." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LandauSymbols.html