Ramanujan tau function

<p>The Ramanujan tau function, studied by <a href="/facts/Srinivasa_Ramanujan/HBPr4bzd">Ramanujan</a> (1916), is the function

τ
        :
        
          N
        
        →
        
          Z
        
      
    
    {\displaystyle \tau :\mathbb {N} \to \mathbb {Z} }
  
 defined by the following identity:
</p>

∑
          
            n
            ≥
            1
          
        
        τ
        (
        n
        )
        
          q
          
            n
          
        
        =
        q
        
          ∏
          
            n
            ≥
            1
          
        
        
          
            (
            
              1
              −
              
                q
                
                  n
                
              
            
            )
          
          
            24
          
        
        =
        q
        ϕ
        (
        q
        
          )
          
            24
          
        
        =
        η
        (
        z
        
          )
          
            24
          
        
        =
        Δ
        (
        z
        )
        ,
      
    
    {\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}\left(1-q^{n}\right)^{24}=q\phi (q)^{24}=\eta (z)^{24}=\Delta (z),}

<p>where 
  
    
      
        q
        =
        exp
        ⁡
        (
        2
        π
        i
        z
        )
      
    
    {\displaystyle q=\exp(2\pi iz)}
  
 with 
  
    
      
        
          I
          m
        
        (
        z
        )
        >
        0
      
    
    {\displaystyle \mathrm {Im} (z)>0}
  
, 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
 is the <a href="/facts/Euler_function/y6lE3VIL">Euler function</a>, 
  
    
      
        η
      
    
    {\displaystyle \eta }
  
 is the <a href="/facts/Dedekind_eta_function/nStBqqmo">Dedekind eta function</a>, and the function 
  
    
      
        Δ
        (
        z
        )
      
    
    {\displaystyle \Delta (z)}
  
 is a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> <a href="/facts/Cusp_form/m0zHUrmc">cusp form</a> of weight 12 and level 1, known as the <a href="/facts/Modular_discriminant/GHRXDknk">discriminant modular form</a> (some authors, notably <a href="/facts/Tom_M._Apostol/S6vGtJWN">Apostol</a>, write 
  
    
      
        Δ
        
          /
        
        (
        2
        π
        
          )
          
            12
          
        
      
    
    {\displaystyle \Delta /(2\pi )^{12}}
  
 instead of 
  
    
      
        Δ
      
    
    {\displaystyle \Delta }
  
). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to <a href="/facts/Ian_G._Macdonald/PoaeNbeg">Ian G. Macdonald</a> was given in Dyson (1972).
</p>

Ramanujan tau function open-in-new

Ramanujan tau function