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# Zeta

 Zeta[s]gives the Riemann zeta function . Zeta[s, a]gives the generalized Riemann zeta function .
• 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.
Generalized (Hurwitz) zeta function:
 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:
Find what fraction of pairs of the first 100 integers are relatively prime:
Compare with a zeta function formula:
Define the xi 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:
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:
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:
Play the real part of the zeta function on the critical line as a sound:
Animate the zeta function on the critical line:
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