Built-in Mathematica Symbol

RiemannSiegelZ

RiemannSiegelZ[t]
gives the Riemann-Siegel function

.

MORE INFORMATION

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.

RiemannSiegelZ can be evaluated to arbitrary numerical precision.

RiemannSiegelZ automatically threads over lists.

EXAMPLES

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Basic Examples (2)

Find a numerical root:

In[1]:=

Out[1]=

Find a numerical root:

Scope (7)

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:

Applications (5)

Plot real and imaginary parts over the complex plane:

View on the branch cut along the imaginary axis:

Find a zero of ReimannSiegelZ 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:

Properties & Relations (2)

Relation to the Riemann zeta function:

Numerically find a root of a transcendental equation:

Possible Issues (2)

A larger setting for $MaxExtraPrecision can be needed:

Machine-number inputs can give high-precision results:

Neat Examples (3)

Recurrence plot of RiemannSiegelZ:

Play RiemannSiegelZ as a sound:

Animate RiemannSiegelZ: