This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# RiemannSiegelZ

 RiemannSiegelZ[t]gives the Riemann-Siegel function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• , where is the Riemann-Siegel theta function, and is the Riemann zeta function.
• for real .
• is an analytic function of except for branch cuts on the imaginary axis running from to .
• For certain special arguments, RiemannSiegelZ automatically evaluates to exact values.
Find a numerical root:
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Find a numerical root:
 Out[1]=
 Out[2]=
 Scope   (7)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Symbolic form of derivatives:
Evaluate derivatives numerically:
 Applications   (5)
Plot real and imaginary parts over the complex plane:
View on the branch cut along the imaginary axis:
Find a zero of using FindRoot:
Or using ZetaZero:
Find several zeros:
Plot curves of vanishing real and imaginary parts of RiemannSiegelZ:
A version of the Riemann hypothesis requires the limit of as to vanish:
Plot double logarithmically the value of the integral:
Calculate a "signal power" of the Riemann zeta function along the critical line:
Plot the difference to the asymptotic value:
Relation to the Riemann zeta function:
Numerically find a root of a transcendental equation:
A larger setting for \$MaxExtraPrecision can be needed:
Machine-number inputs can give high-precision results:
Recurrence plot of RiemannSiegelZ:
Play RiemannSiegelZ as a sound:
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