Big O notation
The Hardy–Littlewood definition

In 1914 Godfrey Harold Hardy and John Edensor Littlewood introduced the new symbol Ω {\displaystyle \Omega }, which is defined as follows:

Thus f (x) = Ω (g (x)) {\displaystyle f(x)=\Omega (g(x))} is the negation of f (x) = o (g (x)) {\displaystyle f(x)=o(g(x))}.

In 1916 the same authors introduced the two new symbols Ω R {\displaystyle \Omega _{R}} and Ω L {\displaystyle \Omega _{L}}, defined as:

These symbols were used by Edmund Landau, with the same meanings, in 1924. After Landau, the notations were never used again exactly thus; Ω R {\displaystyle \Omega _{R}} became Ω + {\displaystyle \Omega _{+}} and Ω L {\displaystyle \Omega _{L}} became Ω − {\displaystyle \Omega _{-}}.

These three symbols Ω, Ω +, Ω − {\displaystyle \Omega,\Omega _{+},\Omega _{-}}, as well as f (x) = Ω ± (g (x)) {\displaystyle f(x)=\Omega _{\pm }(g(x))} (meaning that f (x) = Ω + (g (x)) {\displaystyle f(x)=\Omega _{+}(g(x))} and f (x) = Ω − (g (x)) {\displaystyle f(x)=\Omega _{-}(g(x))} are both satisfied), are now currently used in analytic number theory.
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