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RiemannSiegelTheta

RiemannSiegelTheta[t]
gives the Riemann-Siegel function .
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • for real .
  • arises in the study of the Riemann zeta function on the critical line. It is closely related to the number of zeros of for .
  • is an analytic function of except for branch cuts on the imaginary axis running from to .
  • For certain special arguments, RiemannSiegelTheta automatically evaluates to exact values.
Series expansion at infinity:
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Series expansion at infinity:
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Click for copyable input
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Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
RiemannSiegelTheta threads element-wise over lists:
TraditionalForm formatting:
Series expansion at the origin:
Series expansion at a branch point:
Plot real and imaginary parts over the complex plane:
Show interlacing of the roots of Sin and RiemannSiegelZ.
Numerically find a root of a transcendental equation:
A larger setting for $MaxExtraPrecision can be needed:
Machine-number inputs can give high-precision results:
Riemann surface of RiemannSiegelTheta:
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