BUILT-IN MATHEMATICA SYMBOL

Zeta

Zeta[s]
gives the Riemann zeta function

.

Zeta

gives the generalized Riemann zeta function

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

For , .

, where any term with is excluded.

For , the definition used is .

Zeta[s] has no branch cut discontinuities.

For certain special arguments, Zeta automatically evaluates to exact values.

Zeta can be evaluated to arbitrary numerical precision.

Zeta automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Generalized (Hurwitz) zeta function:

In[1]:=

Out[1]=

In[1]:=

Out[1]=

Generalized (Hurwitz) zeta function:

In[1]:=

Out[1]=

Scope (14)

Evaluate for complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Simple exact values are generated automatically:

Zeta threads element-wise over lists and matrices:

Series expansions at special points:

Evaluate derivatives exactly:

Evaluate derivatives numerically:

TraditionalForm formatting:

Evaluate for complex arguments:

Evaluate to high precision:

Simple exact values are generated automatically:

Use FunctionExpand to generate formulas for other cases:

Formula for a derivative:

TraditionalForm formatting:

Applications (7)

Plot the real part of the zeta function on the critical line:

Plot the real part across the critical strip:

Find a zero of the zeta function:

Find several zeros:

Use ZetaZero:

Find what fraction of pairs of the first 100 integers are relatively prime:

Compare with a zeta function formula:

Define the

function:

Test the Pustyl'nikov form of the Riemann hypothesis:

Plot real and imaginary parts in the vicinity of two very nearby zeros:

Plot the generalized zeta function:

Properties & Relations (7)

Sum involving a zeta function:

Use FullSimplify to prove the functional equation:

The ordinary zeta function is a special case:

In certain cases, FunctionExpand gives formulas in terms of other functions:

Possible Issues (4)

Real and imaginary parts can have very different scales:

Evaluating the imaginary part accurately requires higher internal precision:

Machine-number inputs can give high-precision results:

Giving 0 as an argument does not define the precision required:

Including an accuracy specification gives enough information:

In TraditionalForm,

is not automatically interpreted as the zeta function:

Neat Examples (2)

Play the real part of the zeta function on the critical line as a sound:

Animate the zeta function on the critical line: