Zeta - Maple Help

Zeta

The Riemann Zeta function; the Hurwitz Zeta function

Calling Sequence

 Zeta(z) $\mathrm{\zeta }\left(z\right)$ Zeta(n, z) $\mathrm{\zeta }\left(n,z\right)$ Zeta(n, z, v) $\mathrm{\zeta }\left(n,z,v\right)$

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

$\mathrm{\zeta }\left(z\right)=\sum _{i=1}^{\mathrm{\infty }}\frac{1}{{i}^{z}}$

 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,

$\mathrm{\zeta }\left(n,z\right)=\frac{{ⅆ}^{n}}{ⅆ{z}^{n}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\zeta \left(z\right)$

 • 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 $\mathrm{\zeta }\left(1,\frac{1}{2}\right)$.
 • The optional third parameter v changes the expression of summation to 1/(i+v)^z, so that for Re(z)>1,

$\mathrm{\zeta }\left(n,z,v\right)=\frac{{\partial }^{n}}{\partial {z}^{n}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\sum _{i=0}^{\mathrm{\infty }}\frac{1}{{\left(i+v\right)}^{z}}$

 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

 > $\mathrm{\zeta }\left(2.2\right)$
 ${1.490543257}$ (1)
 > $\mathrm{evalf}\left(\mathrm{\zeta }\left(-1.5+3.5I\right),30\right)$
 ${0.232434139233841813873124398558}{+}{0.173728378830616590886617515292}{}{I}$ (2)
 > $\mathrm{\zeta }\left(1,\frac{1}{2}\right)$
 ${\mathrm{\zeta }}{}\left(\frac{{1}}{{2}}\right){}\left(\frac{{\mathrm{\gamma }}}{{2}}{+}\frac{{\mathrm{ln}}{}\left({8}{}{\mathrm{\pi }}\right)}{{2}}{+}\frac{{\mathrm{\pi }}}{{4}}\right)$ (3)
 > $\mathrm{\zeta }\left(0,2,\frac{1}{2}\right)$
 $\frac{{{\mathrm{\pi }}}^{{2}}}{{2}}$ (4)
 > $\mathrm{\zeta }\left(0,2,s\right)$
 ${\mathrm{\Psi }}{}\left({1}{,}{s}\right)$ (5)
 > $\mathrm{\zeta }\left(3,1.5+0.3I,0.2\right)$
 ${70.20062910}{+}{64.74329586}{}{I}$ (6)
 > $\mathrm{\zeta }\left(3,-1.2+35.3I,0.2+I\right)$
 ${-2.383200150}{×}{{10}}^{{21}}{+}{1.841204211}{×}{{10}}^{{21}}{}{I}$ (7)
 > $\mathrm{sum}\left(\frac{1}{{i}^{7}},i=1..\mathrm{\infty }\right)$
 ${\mathrm{\zeta }}{}\left({7}\right)$ (8)

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

 > $\mathrm{plots}:-\mathrm{complexplot}\left(\mathrm{\zeta }\left(0.5+tI\right),t=0..34,\mathrm{scaling}=\mathrm{constrained},\mathrm{numpoints}=300,\mathrm{labels}=\left["Re","Im"\right]\right)$

References

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