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Zeta

The Riemann Zeta function; the Hurwitz Zeta function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

Zeta(z)

ζz

Zeta(n, z)

ζn,z

Zeta(n, z, v)

ζn,z,v

Parameters

n

-

algebraic expression; understood to be a non-negative integer

z

-

algebraic expression

v

-

algebraic expression; understood not to be a non-positive integer

Description

• 

The Zeta function (zeta function) is defined for Re(z)>1 by

ζz=i=11iz

  

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,

ζn,z=ⅆnⅆznζz

• 

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 ζ1,12.

• 

The optional third parameter v changes the expression of summation to 1/(i+v)^z, so that for Re(z)>1,

ζn,z,v=nzni=01i+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.

Examples

Zeta2.2

1.490543257

(1)

evalfZeta1.5+3.5I,30

0.232434139233841813873124398558+0.173728378830616590886617515292I

(2)

Zeta1,12

ζ12γ2+ln8π2+π4

(3)

Zeta0,2,12

π22

(4)

Zeta0,2,s

Ψ1,s

(5)

Zeta3,1.5+0.3I,0.2

70.20062910+64.74329586I

(6)

Zeta3,1.2+35.3I,0.2+I

−2.3832001501021+1.8412042111021I

(7)

i=1∞1i7

ζ7

(8)

The following plot shows a plot of the Zeta function along the critical line for real values of t from 0 to 34.

plots:-complexplotZeta0.5+tI,t=0..34,scaling=constrained,numpoints=300,labels=Re,Im

References

  

Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953. Vol. 1.

See Also

initialfunctions

JacobiZeta

PolynomialTools[Hurwitz]