Liouville function

The Liouville lambda function, denoted by λ(n) and named after <a href="/facts/Joseph_Liouville/U22o7wtK">Joseph Liouville</a>, is an important <a href="/facts/Arithmetic_function/hULxwsgm">arithmetic function</a>. 
Its value is +1 if n is the product of an even number of <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>, and −1 if it is the product of an odd number of primes. 
Explicitly, the <a href="/facts/Fundamental_theorem_of_arithmetic/EzHUwlXX">fundamental theorem of arithmetic</a> states that any positive <a href="/facts/Integer/PUwwrawx">integer</a> n can be represented uniquely as a product of powers of primes: n = p1a1 ⋯ pkak, where p1 < p2 < ... < pk are primes and the aj are positive integers. (1 is given by the empty product.) The <a href="/facts/Prime_omega_function/obfFNVEB">prime omega functions</a> count the number of primes, with (Ω) or without (ω) multiplicity:

ω
        (
        n
        )
        =
        k
        ,
      
    
    {\displaystyle \omega (n)=k,}

Ω
        (
        n
        )
        =
        
          a
          
            1
          
        
        +
        
          a
          
            2
          
        
        +
        ⋯
        +
        
          a
          
            k
          
        
        .
      
    
    {\displaystyle \Omega (n)=a_{1}+a_{2}+\cdots +a_{k}.}

λ(n) is defined by the formula

λ
        (
        n
        )
        =
        (
        −
        1
        
          )
          
            Ω
            (
            n
            )
          
        
      
    
    {\displaystyle \lambda (n)=(-1)^{\Omega (n)}}

(sequence A008836 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>).
λ is <a href="/facts/Multiplicative_function/4kfolbBl">completely multiplicative</a> since Ω(n) is completely <a href="/facts/Additive_function/J5XLJSz4">additive</a>, 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 <a href="/facts/M%25C3%25B6bius_function/geuvVfqS">Möbius function</a> μ(n). Write n as n = a2b, where b is <a href="/facts/Squarefree/R3qR9hP1">squarefree</a>, i.e., ω(b) = Ω(b). Then

λ
        (
        n
        )
        =
        μ
        (
        b
        )
        .
      
    
    {\displaystyle \lambda (n)=\mu (b).}

The sum of the Liouville function over the <a href="/facts/Divisor/DKNa2GvC">divisors</a> of n is the <a href="/facts/Indicator_function/QEuh04NM">characteristic function</a> of the <a href="/facts/Square_(algebra)/Jzpq0R0z">squares</a>:

∑
          
            d
            
              |
            
            n
          
        
        λ
        (
        d
        )
        =
        
          
            {
            
              
                
                  1
                
                
                  
                    if 
                  
                  n
                  
                     is a perfect square,
                  
                
              
              
                
                  0
                
                
                  
                    otherwise.
                  
                
              
            
            
          
        
      
    
    {\displaystyle \sum _{d|n}\lambda (d)={\begin{cases}1&{\text{if }}n{\text{ is a perfect square,}}\\0&{\text{otherwise.}}\end{cases}}}

<a href="/facts/M%25C3%25B6bius_inversion_formula/xDnSrPp6">Möbius inversion</a> of this formula yields

λ
        (
        n
        )
        =
        
          ∑
          
            
              d
              
                2
              
            
            
              |
            
            n
          
        
        μ
        
          (
          
            
              n
              
                d
                
                  2
                
              
            
          
          )
        
        .
      
    
    {\displaystyle \lambda (n)=\sum _{d^{2}|n}\mu \left({\frac {n}{d^{2}}}\right).}

The <a href="/facts/Dirichlet_inverse/DjuhDu1L">Dirichlet inverse</a> of Liouville function is the absolute value of the Möbius function, λ–1(n) = |μ(n)| = μ2(n), the characteristic function of the squarefree integers.

Liouville function open-in-new

Liouville function