The Riemann Zeta function; the Hurwitz Zeta function
Zeta(n, z, v)
algebraic expression; understood to be a non-negative integer
algebraic expression; understood not to be a non-positive integer
The Zeta function (zeta function) is defined for Re(z)>1 by
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,
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,
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.
The following plot shows a plot of the Zeta function along the critical line for real values of t from 0 to 34.
Erdelyi, A. Higher Transcendental Functions. McGraw-Hill, 1953. Vol. 1.
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