Zeta
The Riemann Zeta function; the Hurwitz Zeta function
Calling Sequence
Parameters
Description
Examples
References
Zeta(z)
ζ⁡z
Zeta(n, z)
ζ⁡n,z
Zeta(n, z, v)
ζ⁡n,z,v
n
-
algebraic expression; understood to be a non-negative integer
z
algebraic expression
v
algebraic expression; understood not to be a non-positive integer
The Zeta function (zeta function) is defined for Re(z)>1 by
ζ⁡z=∑i=1∞⁡1iz
and is extended to the rest of the complex plane (except for the point z=1) by analytic continuation. The point z=1 is a simple pole.
The call Zeta(n, z) gives the nth derivative of the Zeta function,
ζ⁡n,z=ⅆnⅆznζ⁡z
You can enter the command Zeta using either the 1-D or 2-D calling sequence. For example, Zeta(1, 1/2) is equivalent to ζ⁡1,12.
The optional third parameter v changes the expression of summation to 1/(i+v)^z, so that for Re(z)>1,
ζ⁡n,z,v=∂n∂zn∑i=0∞⁡1i+vz
and, again, this is extended to the complex plane less the point 1 by analytic continuation. The point z=1 is a simple pole for the function Zeta(0, z, v).
The third parameter, v, can be any complex number which is not a non-positive integer.
The function Zeta(0, z, v) is often called the Hurwitz Zeta function or the Generalized Zeta function.
Zeta⁡2.2
1.490543257
evalf⁡Zeta⁡−1.5+3.5⁢I,30
0.232434139233841813873124398558+0.173728378830616590886617515292⁢I
Zeta⁡1,12
ζ⁡12⁢γ2+ln⁡8⁢π2+π4
Zeta⁡0,2,12
π22
Zeta⁡0,2,s
Ψ⁡1,s
Zeta⁡3,1.5+0.3⁢I,0.2
70.20062910+64.74329586⁢I
Zeta⁡3,−1.2+35.3⁢I,0.2+I
−2.383200150×1021+1.841204211×1021⁢I
∑i=1∞1i7
ζ⁡7
The following plot shows a plot of the Zeta function along the critical line for real values of t from 0 to 34.
plots:-complexplot⁡Zeta⁡0.5+t⁢I,t=0..34,scaling=constrained,numpoints=300,labels=Re,Im
Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953. Vol. 1.
See Also
initialfunctions
JacobiZeta
MultiZeta
PolynomialTools[Hurwitz]
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