Dedekind psi function

<h2 id="higher-orders">Higher orders</h2>
<p>The generalization to higher orders via ratios of <a href="/facts/Jordan%27s_totient_function/SpKAT70W">Jordan's totient</a> is
</p>

ψ
          
            k
          
        
        (
        n
        )
        =
        
          
            
              
                J
                
                  2
                  k
                
              
              (
              n
              )
            
            
              
                J
                
                  k
                
              
              (
              n
              )
            
          
        
      
    
    {\displaystyle \psi _{k}(n)={\frac {J_{2k}(n)}{J_{k}(n)}}}

<p>with Dirichlet series
</p>

∑
          
            n
            ≥
            1
          
        
        
          
            
              
                ψ
                
                  k
                
              
              (
              n
              )
            
            
              n
              
                s
              
            
          
        
        =
        
          
            
              ζ
              (
              s
              )
              ζ
              (
              s
              −
              k
              )
            
            
              ζ
              (
              2
              s
              )
            
          
        
      
    
    {\displaystyle \sum _{n\geq 1}{\frac {\psi _{k}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-k)}{\zeta (2s)}}}
  
.
<p>It is also the <a href="/facts/Dirichlet_convolution/DjuhDu1L">Dirichlet convolution</a> of a power and the square
of the <a href="/facts/M%C3%B6bius_function/geuvVfqS">Möbius function</a>,
</p>

ψ
          
            k
          
        
        (
        n
        )
        =
        
          n
          
            k
          
        
        ∗
        
          μ
          
            2
          
        
        (
        n
        )
      
    
    {\displaystyle \psi _{k}(n)=n^{k}*\mu ^{2}(n)}
  
.
<p>If
</p>

ϵ
          
            2
          
        
        =
        1
        ,
        0
        ,
        0
        ,
        1
        ,
        0
        ,
        0
        ,
        0
        ,
        0
        ,
        1
        ,
        0
        ,
        0
        ,
        0
        ,
        0
        ,
        0
        ,
        0
        ,
        1
        ,
        0
        ,
        0
        ,
        0
        …
      
    
    {\displaystyle \epsilon _{2}=1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0\ldots }

<p>is the <a href="/facts/Indicator_function/QEuh04NM">characteristic function</a> of the squares, another Dirichlet convolution
leads to the generalized <a href="/facts/Divisor_function/rzvm1jw2">σ-function</a>,
</p>

ϵ
          
            2
          
        
        (
        n
        )
        ∗
        
          ψ
          
            k
          
        
        (
        n
        )
        =
        
          σ
          
            k
          
        
        (
        n
        )
      
    
    {\displaystyle \epsilon _{2}(n)*\psi _{k}(n)=\sigma _{k}(n)}
  
.

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/DedekindFunction.html">"Dedekind Function"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Goro_Shimura/pRxwqhyF">Goro Shimura</a> (1971). <i>Introduction to the Arithmetic Theory of Automorphic Functions</i>. Princeton. (page 25, equation (1))</li>
<li>Mathar, Richard J. (2011). "Survey of Dirichlet series of multiplicative arithmetic functions". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1106.4038">1106.4038</a> [<a href="https://arxiv.org/archive/math.NT">math.NT</a>]. Section 3.13.2</li>
<li><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A065958 is ψ2, <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A065959 is ψ3, and <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A065960 is ψ4</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Leonard Eugene Dickson "History of the Theory Of Numbers", Vol. 1, p. 123, Chelsea Publishing 1952. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Journal für die reine und angewandte Mathematik, vol. 83, 1877, p. 288. Cf. H. Weber, Elliptische Functionen, 1901, 244-5; ed. 2, 1008 (Algebra III), 234-5 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Dedekind psi function open-in-new

Dedekind psi function