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Liouville lambda function

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The Liouville lambda function , where is the number of prime factors function (sometimes called "big omega"), tells whether has an odd or even number of prime factors, not necessarily distinct. The Liouville lambda function is sometimes denoted by . The function is completely multiplicative, meaning that regardless of whether or not.

For example, because 26 has two prime factors (2 and 13) and so . because 27 has three prime factors (3 thrice) and so . With we exhibit that the function is completely multiplicative, since . See A008836 for more values.

Theorem. The Liouville lambda function is completely multiplicative. Given , the equality holds even when .
Proof. Remember that whether or not (whereas with the number of distinct prime factors function this would not be the case). Therefore, if is even and is also even, then so is and thus . If is even but is odd, then is odd and thus . But if both and are odd, then is even and thus . ENDOFPROOFMARK

As the Mertens function is to the Möbius function, so is the Liouville summatory function (see A002819). See Pólya's conjecture.