Riemann–Siegel theta function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Riemann–Siegel theta function is defined in terms of the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> as

θ
        (
        t
        )
        =
        arg
        ⁡
        
          (
          
            Γ
            
              (
              
                
                  
                    1
                    4
                  
                
                +
                
                  
                    
                      i
                      t
                    
                    2
                  
                
              
              )
            
          
          )
        
        −
        
          
            
              log
              ⁡
              π
            
            2
          
        
        t
      
    
    {\displaystyle \theta (t)=\arg \left(\Gamma \left({\frac {1}{4}}+{\frac {it}{2}}\right)\right)-{\frac {\log \pi }{2}}t}

for real values of t. Here the <a href="/facts/Argument_(complex_analysis)/JSjfxb5U">argument</a> is chosen in such a way that a continuous function is obtained and 
 
 
 
 θ
 (
 0
 )
 =
 0
 
 
 {\displaystyle \theta (0)=0}
 
 holds, i.e., in the same way that the <a href="/facts/Principal_branch/XrN2gNon">principal branch</a> of the <a href="/facts/Gamma_function/SjGQcnyw">log-gamma</a> function is defined.
It has an <a href="/facts/Asymptotic_expansion/4W79BNve">asymptotic expansion</a>

θ
        (
        t
        )
        ∼
        
          
            t
            2
          
        
        log
        ⁡
        
          
            t
            
              2
              π
            
          
        
        −
        
          
            t
            2
          
        
        −
        
          
            π
            8
          
        
        +
        
          
            1
            
              48
              t
            
          
        
        +
        
          
            7
            
              5760
              
                t
                
                  3
                
              
            
          
        
        +
        ⋯
      
    
    {\displaystyle \theta (t)\sim {\frac {t}{2}}\log {\frac {t}{2\pi }}-{\frac {t}{2}}-{\frac {\pi }{8}}+{\frac {1}{48t}}+{\frac {7}{5760t^{3}}}+\cdots }

which is not convergent, but whose first few terms give a good approximation for 
 
 
 
 t
 ≫
 1
 
 
 {\displaystyle t\gg 1}
 
. Its Taylor-series at 0 which converges for 
 
 
 
 
 |
 
 t
 
 |
 
 <
 1
 
 /
 
 2
 
 
 {\displaystyle |t|<1/2}
 
 is

θ
        (
        t
        )
        =
        −
        
          
            t
            2
          
        
        log
        ⁡
        π
        +
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          
            
              (
              −
              1
              
                )
                
                  k
                
              
              
                ψ
                
                  (
                  2
                  k
                  )
                
              
              
                (
                
                  
                    1
                    4
                  
                
                )
              
            
            
              (
              2
              k
              +
              1
              )
              !
            
          
        
        
          
            (
            
              
                t
                2
              
            
            )
          
          
            2
            k
            +
            1
          
        
      
    
    {\displaystyle \theta (t)=-{\frac {t}{2}}\log \pi +\sum _{k=0}^{\infty }{\frac {(-1)^{k}\psi ^{(2k)}\left({\frac {1}{4}}\right)}{(2k+1)!}}\left({\frac {t}{2}}\right)^{2k+1}}

where 
 
 
 
 
 ψ
 
 (
 2
 k
 )
 
 
 
 
 {\displaystyle \psi ^{(2k)}}
 
 denotes the <a href="/facts/Polygamma_function/IF6W68aL">polygamma function</a> of order 
 
 
 
 2
 k
 
 
 {\displaystyle 2k}
 
.
The Riemann–Siegel theta function is of interest in studying the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>, since it can rotate the Riemann zeta function such that it becomes the totally real valued <a href="/facts/Z_function/uaeSLTUv">Z function</a> on the <a href="/facts/Critical_line_(mathematics)/Td8Jq5r0">critical line</a> 
 
 
 
 s
 =
 1
 
 /
 
 2
 +
 i
 t
 
 
 {\displaystyle s=1/2+it}
 
 .

Riemann–Siegel theta function open-in-new

Riemann–Siegel theta function