Liouville function
The Liouville Lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product of an even number of prime numbers, and −1 if it is the product of an odd number of primes. Explicitly, the fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p1 < p2 < ... < pk are primes and the aj are positive integers. (1 is given by the empty product.) The prime omega functions count the number of primes, with (Ω) or without (ω) multiplicity: ω(n) = k,Ω(n) = a1 + a2 + ... + ak. λ(n) is defined by the formula (sequence in the OEIS). λ is completely multiplicative since Ω(n) is completely additive, i.e.: Ω(ab) = Ω(a) + Ω(b). Since 1 has no prime factors, Ω(1) = 0 so λ(1) = 1. It is related to the Möbius function μ(n). Write n as n = a2b where b is squarefree, i.e., ω(b) = Ω(b). Then The sum of the Liouville function over the divisors of n is the characteristic function of the squares: Möbius inversion of this formula yields The Dirichlet inverse of Liouville function is the absolute value of the Möbius function, the characteristic function of the squarefree integers. We also have that . (Wikipedia).
