Editing Riemann–Siegel theta function (section)

==Theta as a function of a complex variable==
We have an infinite series expression for the [[Gamma function#The log-gamma function|log-gamma]] function

:<math>\log \Gamma \left(z\right) = -\gamma z -\log z  
+ \sum_{n=1}^\infty 
\left(\frac{z}{n} - \log \left(1+\frac{z}{n}\right)\right),</math>

where ''γ'' is [[Euler–Mascheroni constant|Euler's constant]]. Substituting <math>(2it+1)/4</math> for ''z'' and taking the imaginary part termwise gives the following series for ''θ''(''t'')

:<math>\theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t + \sum_{n=1}^\infty \left(\frac{t}{2n} - \arctan\left(\frac{2t}{4n+1}\right)\right).</math>

For values with imaginary part between −1 and 1, the arctangent function is [[holomorphic function|holomorphic]], and it is easily seen that the series converges uniformly on compact sets in the region with imaginary part between −1/2 and 1/2, leading to a holomorphic function on this domain. It follows that the [[Z function]] is also holomorphic in this region, which is the critical strip.

We may use the identities

:<math>\arg z = \frac{\log z - \log\bar z}{2i}\quad\text{and}\quad\overline{\Gamma(z)}=\Gamma(\bar z)</math>

to obtain the closed-form expression

:<math>\theta(t) = \frac{\log\Gamma\left(\frac{2it+1}{4}\right)-\log\Gamma\left(\frac{-2it+1}{4}\right)}{2i} - \frac{\log \pi}{2} t=- \frac{i}{2} \left( \ln \Gamma \left( \frac{1}{4} + \frac{i t}{2} \right) -
\ln \Gamma \left( \frac{1}{4} - \frac{i t}{2} \right) \right) - \frac{\ln (\pi) t}{2}</math>

which extends our original definition to a holomorphic function of ''t''.  Since the principal branch of log Γ has a single branch cut along the negative real axis, ''θ''(''t'') in this definition inherits branch cuts along the imaginary axis above ''i''/2 and below&nbsp;−''i''/2.

{|  style="text-align:center"
|+ '''Riemann–Siegel theta function in the complex plane'''
|[[Image:Riemann Siegel Theta 1.jpg|1000x140px|none]]
|[[Image:Riemann Siegel Theta 2.jpg|1000x140px|none]]
|[[Image:Riemann Siegel Theta 3.jpg|1000x140px|none]]
|-
|<math>
 -1 < \Re(t) < 1
</math>
|<math>
 -5 < \Re(t) < 5
</math>
|<math>
 -40 < \Re(t) < 40
</math>
|}