Dedekind psi function

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the Dedekind psi function is the <a href="/facts/Multiplicative_function/4kfolbBl">multiplicative function</a> on the positive integers defined by

ψ
        (
        n
        )
        =
        n
        
          ∏
          
            p
            
              |
            
            n
          
        
        
          (
          
            1
            +
            
              
                1
                p
              
            
          
          )
        
        ,
      
    
    {\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right),}

where the product is taken over all primes 
 
 
 
 p
 
 
 {\displaystyle p}
 
 dividing 
 
 
 
 n
 .
 
 
 {\displaystyle n.}
 
 (By convention, 
 
 
 
 ψ
 (
 1
 )
 
 
 {\displaystyle \psi (1)}
 
, which is the <a href="/facts/Empty_product/mj7nhGeD">empty product</a>, has value 1.) The function was introduced by <a href="/facts/Richard_Dedekind/CLkWDhY3">Richard Dedekind</a> in connection with <a href="/facts/Modular_function/PcnbYR1f">modular functions</a>.
The value of 
 
 
 
 ψ
 (
 n
 )
 
 
 {\displaystyle \psi (n)}
 
 for the first few integers 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is:

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... (sequence A001615 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>).
The function 
 
 
 
 ψ
 (
 n
 )
 
 
 {\displaystyle \psi (n)}
 
 is greater than 
 
 
 
 n
 
 
 {\displaystyle n}
 
 for all 
 
 
 
 n
 
 
 {\displaystyle n}
 
 greater than 1, and is even for all 
 
 
 
 n
 
 
 {\displaystyle n}
 
 greater than 2. If 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is a <a href="/facts/Square-free_number/R3qR9hP1">square-free number</a> then 
 
 
 
 ψ
 (
 n
 )
 =
 σ
 (
 n
 )
 
 
 {\displaystyle \psi (n)=\sigma (n)}
 
, where 
 
 
 
 σ
 (
 n
 )
 
 
 {\displaystyle \sigma (n)}
 
 is the <a href="/facts/Divisor_function/rzvm1jw2">sum-of-divisors function</a>.
The 
 
 
 
 ψ
 
 
 {\displaystyle \psi }
 
 function can also be defined by setting 
 
 
 
 ψ
 (
 
 p
 
 n
 
 
 )
 =
 (
 p
 +
 1
 )
 
 p
 
 n
 −
 1
 
 
 
 
 {\displaystyle \psi (p^{n})=(p+1)p^{n-1}}
 
 for powers of any prime 
 
 
 
 p
 
 
 {\displaystyle p}
 
, and then extending the definition to all integers by multiplicativity. This also leads to a proof of the <a href="/facts/Generating_function/K7UVjnjL">generating function</a> in terms of the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>, which is

∑
        
          
            
              ψ
              (
              n
              )
            
            
              n
              
                s
              
            
          
        
        =
        
          
            
              ζ
              (
              s
              )
              ζ
              (
              s
              −
              1
              )
            
            
              ζ
              (
              2
              s
              )
            
          
        
        .
      
    
    {\displaystyle \sum {\frac {\psi (n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-1)}{\zeta (2s)}}.}

This is also a consequence of the fact that we can write as a <a href="/facts/Dirichlet_convolution/DjuhDu1L">Dirichlet convolution</a> of 
 
 
 
 ψ
 =
 
 I
 d
 
 ∗
 
 |
 
 μ
 
 |
 
 
 
 {\displaystyle \psi =\mathrm {Id} *|\mu |}
 
.
There is an additive definition of the psi function as well. Quoting from Dickson,

R. Dedekind proved that, if 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is decomposed in every way into a product 
 
 
 
 a
 b
 
 
 {\displaystyle ab}
 
 and if 
 
 
 
 e
 
 
 {\displaystyle e}
 
 is the g.c.d. of 
 
 
 
 a
 ,
 b
 
 
 {\displaystyle a,b}
 
 then

∑
          
            a
          
        
        (
        a
        
          /
        
        e
        )
        φ
        (
        e
        )
        =
        n
        
          ∏
          
            p
            
              |
            
            n
          
        
        
          (
          
            1
            +
            
              
                1
                p
              
            
          
          )
        
      
    
    {\displaystyle \sum _{a}(a/e)\varphi (e)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right)}

where 
 
 
 
 a
 
 
 {\displaystyle a}
 
 ranges over all divisors of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 and 
 
 
 
 p
 
 
 {\displaystyle p}
 
 over the prime divisors of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 and 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 is the <a href="/facts/Totient_function/mIQwXYzj">totient function</a>.

Dedekind psi function open-in-new

Dedekind psi function