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Mathematica > Mathematics and Algorithms > Number Theory > Prime Numbers >

Built-in Mathematica Symbol

LiouvilleLambda

LiouvilleLambda[n]
gives the Liouville function

Integer mathematical function, suitable for both symbolic and numerical manipulation.

For a number n=u p₁^k₁ p_m^k_m with u a unit and p_i primes, LiouvilleLambda[n] returns (-1)^k₁++k_m.

LiouvilleLambda[n, GaussianIntegers->True] is defined using factorization over Gaussian integers.

Liouville's function gives

where

is the number of prime factors counting multiplicity:

Plot the sequence:

Liouville's function gives

where

is the number of prime factors counting multiplicity:

Plot the sequence:

Evaluate for large arguments:

LiouvilleLambda works over Gaussian integers:

Compute its Dirichlet transform:

LiouvilleLambda threads element-wise over lists:

Counting Gaussian integer prime factors:

LiouvilleLambda is completely multiplicative: