Shintani zeta function

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        P
        (
        
          x
        
        )
      
    
    {\displaystyle P(\mathbf {x} )}
  
 be a polynomial in the variables 
  
    
      
        
          x
        
        =
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            r
          
        
        )
      
    
    {\displaystyle \mathbf {x} =(x_{1},\dots ,x_{r})}
  
 with real coefficients such that 
  
    
      
        P
        (
        
          x
        
        )
      
    
    {\displaystyle P(\mathbf {x} )}
  
 is a product of linear polynomials with positive coefficients, that is, 
  
    
      
        P
        (
        
          x
        
        )
        =
        
          P
          
            1
          
        
        (
        
          x
        
        )
        
          P
          
            2
          
        
        (
        
          x
        
        )
        ⋯
        
          P
          
            k
          
        
        (
        
          x
        
        )
      
    
    {\displaystyle P(\mathbf {x} )=P_{1}(\mathbf {x} )P_{2}(\mathbf {x} )\cdots P_{k}(\mathbf {x} )}
  
, where 
  
    
      
        
          P
          
            i
          
        
        (
        
          x
        
        )
        =
        
          a
          
            i
            1
          
        
        
          x
          
            1
          
        
        +
        
          a
          
            i
            2
          
        
        
          x
          
            2
          
        
        +
        ⋯
        +
        
          a
          
            i
            r
          
        
        
          x
          
            r
          
        
        +
        
          b
          
            i
          
        
        ,
      
    
    {\displaystyle P_{i}(\mathbf {x} )=a_{i1}x_{1}+a_{i2}x_{2}+\cdots +a_{ir}x_{r}+b_{i},}
  
where 
  
    
      
        
          a
          
            i
            j
          
        
        >
        0
      
    
    {\displaystyle a_{ij}>0}
  
, 
  
    
      
        
          b
          
            i
          
        
        >
        0
      
    
    {\displaystyle b_{i}>0}
  
 and 
  
    
      
        k
        =
        deg
        ⁡
        P
      
    
    {\displaystyle k=\deg P}
  
. The <i>Shintani zeta function</i> in the variable 
  
    
      
        s
      
    
    {\displaystyle s}
  
 is given by (the meromorphic continuation of)
  
    
      
        ζ
        (
        P
        ;
        s
        )
        =
        
          ∑
          
            
              x
              
                1
              
            
            ,
            …
            ,
            
              x
              
                r
              
            
            =
            1
          
          
            ∞
          
        
        
          
            1
            
              P
              (
              
                x
              
              
                )
                
                  s
                
              
            
          
        
        .
      
    
    {\displaystyle \zeta (P;s)=\sum _{x_{1},\dots ,x_{r}=1}^{\infty }{\frac {1}{P(\mathbf {x} )^{s}}}.}
  
 
</p>
<h2 id="the-multi-variable-version">The multi-variable version</h2>
<p>The definition of Shintani zeta function has a straightforward generalization to a zeta function in several variables 
  
    
      
        (
        
          s
          
            1
          
        
        ,
        …
        ,
        
          s
          
            k
          
        
        )
      
    
    {\displaystyle (s_{1},\dots ,s_{k})}
  
 given by
  
    
      
        
          ∑
          
            
              x
              
                1
              
            
            ,
            …
            ,
            
              x
              
                r
              
            
            =
            1
          
          
            ∞
          
        
        
          
            1
            
              
                P
                
                  1
                
              
              (
              
                x
              
              
                )
                
                  
                    s
                    
                      1
                    
                  
                
              
              ⋯
              
                P
                
                  k
                
              
              (
              
                x
              
              
                )
                
                  
                    s
                    
                      k
                    
                  
                
              
            
          
        
        .
      
    
    {\displaystyle \sum _{x_{1},\dots ,x_{r}=1}^{\infty }{\frac {1}{P_{1}(\mathbf {x} )^{s_{1}}\cdots P_{k}(\mathbf {x} )^{s_{k}}}}.}
  
The special case when <i>k</i> = 1 is the <a href="/facts/Barnes_zeta_function/6k542ToH">Barnes zeta function</a>.
</p>
<h2 id="relation-to-witten-zeta-functions">Relation to Witten zeta functions</h2>
<p>Just like Shintani zeta functions, <a href="/facts/Witten_zeta_function/r2sj3X0m">Witten zeta functions</a> are defined by polynomials which are products of linear forms with non-negative coefficients. Witten zeta functions are however not special cases of Shintani zeta functions because in Witten zeta functions the linear forms are allowed to have some coefficients equal to zero.  For example, the polynomial 
  
    
      
        (
        x
        +
        1
        )
        (
        y
        +
        1
        )
        (
        x
        +
        y
        +
        2
        )
        
          /
        
        2
      
    
    {\displaystyle (x+1)(y+1)(x+y+2)/2}
  
 defines the Witten zeta function of 
  
    
      
        S
        U
        (
        3
        )
      
    
    {\displaystyle SU(3)}
  
 but the linear form 
  
    
      
        x
        +
        1
      
    
    {\displaystyle x+1}
  
 has 
  
    
      
        y
      
    
    {\displaystyle y}
  
-coefficient equal to zero. 
</p>

<ul><li>Hida, Haruzo (1993), <i>Elementary theory of L-functions and Eisenstein series</i>, London Mathematical Society Student Texts, vol. 26, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-43411-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1216135">1216135</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0942.11024">0942.11024</a></li>
<li>Shintani, Takuro (1976), "On evaluation of zeta functions of totally real algebraic number fields at non-positive integers", <i>Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics</i>, 23 (2): 393–417, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0040-8980">0040-8980</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0427231">0427231</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0349.12007">0349.12007</a></li></ul>

Shintani zeta function open-in-new

Shintani zeta function