Riesz function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Riesz function is an <a href="/facts/Entire_function/XNluKzu7">entire function</a> defined by <a href="/facts/Marcel_Riesz/9jWL9fQz">Marcel Riesz</a> in connection with the <a href="/facts/Riemann_hypothesis/Td8Jq5r0">Riemann hypothesis</a>, by means of the <a href="/facts/Power_series/vYh6sTps">power series</a>

R
            i
            e
            s
            z
          
        
        (
        x
        )
        =
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            
              (
              −
              1
              
                )
                
                  k
                  −
                  1
                
              
              
                x
                
                  k
                
              
            
            
              (
              k
              −
              1
              )
              !
              ζ
              (
              2
              k
              )
            
          
        
        =
        x
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            
              μ
              (
              n
              )
            
            
              n
              
                2
              
            
          
        
        exp
        ⁡
        
          (
          
            
              
                −
                x
              
              
                n
                
                  2
                
              
            
          
          )
        
        .
      
    
    {\displaystyle {\rm {Riesz}}(x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}x^{k}}{(k-1)!\zeta (2k)}}=x\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{2}}}\exp \left({\frac {-x}{n^{2}}}\right).}

If we set 
 
 
 
 F
 (
 x
 )
 =
 
 
 1
 2
 
 
 
 
 R
 i
 e
 s
 z
 
 
 (
 4
 
 π
 
 2
 
 
 x
 )
 
 
 {\displaystyle F(x)={\frac {1}{2}}{\rm {Riesz}}(4\pi ^{2}x)}
 
 we may define it in terms of the coefficients of the <a href="/facts/Laurent_series/sSMevFU8">Laurent series</a> development of the hyperbolic (or equivalently, the ordinary) cotangent around zero. If

x
            2
          
        
        coth
        ⁡
        
          
            x
            2
          
        
        =
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          c
          
            n
          
        
        
          x
          
            n
          
        
        =
        1
        +
        
          
            1
            12
          
        
        
          x
          
            2
          
        
        −
        
          
            1
            720
          
        
        
          x
          
            4
          
        
        +
        ⋯
      
    
    {\displaystyle {\frac {x}{2}}\coth {\frac {x}{2}}=\sum _{n=0}^{\infty }c_{n}x^{n}=1+{\frac {1}{12}}x^{2}-{\frac {1}{720}}x^{4}+\cdots }

then 
 
 
 
 F
 
 
 {\displaystyle F}
 
 may be defined as

F
        (
        x
        )
        =
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            
              x
              
                k
              
            
            
              
                c
                
                  2
                  k
                
              
              (
              k
              −
              1
              )
              !
            
          
        
        =
        12
        x
        −
        720
        
          x
          
            2
          
        
        +
        15120
        
          x
          
            3
          
        
        −
        ⋯
      
    
    {\displaystyle F(x)=\sum _{k=1}^{\infty }{\frac {x^{k}}{c_{2k}(k-1)!}}=12x-720x^{2}+15120x^{3}-\cdots }

The values of 
 
 
 
 ζ
 (
 2
 k
 )
 
 
 {\displaystyle \zeta (2k)}
 
 approach one for increasing k, and comparing the series for the Riesz function with that for 
 
 
 
 x
 exp
 ⁡
 (
 −
 x
 )
 
 
 {\displaystyle x\exp(-x)}
 
 shows that it defines an entire function. Alternatively, F may be defined as

F
        (
        x
        )
        =
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            
              
                k
                
                  
                    
                      k
                      +
                      1
                    
                    ¯
                  
                
              
              
                x
                
                  k
                
              
            
            
              B
              
                2
                k
              
            
          
        
        .
         
      
    
    {\displaystyle F(x)=\sum _{k=1}^{\infty }{\frac {k^{\overline {k+1}}x^{k}}{B_{2k}}}.\ }

n
 
 
 k
 ¯
 
 
 
 
 
 {\displaystyle n^{\overline {k}}}
 
 denotes the <a href="/facts/Pochhammer_symbol/sAxnmGoC">rising factorial power</a> in the notation of <a href="/facts/D._E._Knuth/GozNzDM6">D. E. Knuth</a> and the number 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 are the <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli number</a>. The series is one of alternating terms and the function quickly tends to minus infinity for increasingly negative values of 
 
 
 
 x
 
 
 {\displaystyle x}
 
. Positive values of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 are more interesting and delicate.

Riesz function open-in-new

Riesz function