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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:
Click for copyable input
Find a numerical root:
Click for copyable input
Click for copyable input
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
RiemannSiegelZ threads element-wise over lists:
Symbolic form of derivatives:
Evaluate derivatives numerically:
TraditionalForm formatting:
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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