Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how

LiouvilleLambda

LiouvilleLambda[n]

gives the Liouville function .

Details and Options

LiouvilleLambda is also known as Liouville function.
Integer mathematical function, suitable for both symbolic and numerical manipulation.
LiouvilleLambda[n] gives 1 whenever the number of prime factors counting multiplicity of n is even and -1 otherwise.
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.

With the setting GaussianIntegers->True, LiouvilleLambda is defined using factorization over Gaussian integers.
LiouvilleLambda[m+In] automatically works over Gaussian integers.

Examples

open allclose all

Basic Examples (2)

Compute the Liouville function at and :

Plot the LiouvilleLambda sequence for the first 20 numbers:

Scope (8)

Numerical Evaluation (4)

LiouvilleLambda works over integers:

Gaussian integers:

Compute for large integers:

LiouvilleLambda threads over lists:

Symbolic Manipulation (4)

TraditionalForm formatting:

Reduce expressions:

Solve equations:

DirichletTransform of LiouvilleLambda:

Equivalently:

Options (1)

GaussianIntegers (1)

Compute LiouvilleLambda over integers:

Gaussian integers:

Applications (5)

Basic Applications (2)

Highlight numbers for which Liouville's function is in blue and for which it is in red:

Histogram of the cumulative values of LiouvilleLambda:

Number Theory (3)

LiouvilleLambda and MoebiusMu are related by the equation :

Use LiouvilleLambda to compute MoebiusMu:

Plot the summatory function :

Properties & Relations (5)

LiouvilleLambda is a completely multiplicative function:

LiouvilleLambda gives when the argument is a product of an even number of primes:

Otherwise, it gives :

Use FactorInteger to compute LiouvilleLambda:

Use PrimeOmega to compute LiouvilleLambda:

gives for a perfect square n and otherwise:

Neat Examples (3)

Plot LiouvilleLambda for the sum of two squares:

Plot the arguments of the Fourier transform of LiouvilleLambda:

Plot the Ulam spiral of LiouvilleLambda:

Top