# zeta

Riemann zeta function

## Description

example

zeta(z) evaluates the Riemann zeta function at the elements of z, where z is a numeric or symbolic input.

example

zeta(n,z) returns the nth derivative of zeta(z).

## Examples

### Find Riemann Zeta Function for Numeric and Symbolic Inputs

Find the Riemann zeta function for numeric inputs.

zeta([0.7 i 4 11/3])
ans =
-2.7784 + 0.0000i   0.0033 - 0.4182i   1.0823 + 0.0000i   1.1094 + 0.0000i

Find the Riemann zeta function symbolically by converting the inputs to symbolic objects using sym. The zeta function returns exact results.

zeta(sym([0.7 i 4 11/3]))
ans =
[ zeta(7/10), zeta(1i), pi^4/90, zeta(11/3)]

zeta returns unevaluated function calls for symbolic inputs that do not have results implemented. The implemented results are listed in Algorithms.

Find the Riemann zeta function for a matrix of symbolic expressions.

syms x y
Z = zeta([x sin(x); 8*x/11 x + y])
Z =
[        zeta(x), zeta(sin(x))]
[ zeta((8*x)/11),  zeta(x + y)]

### Find Riemann Zeta Function for Large Inputs

For values of |z|>1000, zeta(z) might return an unevaluated function call. Use expand to force zeta to evaluate the function call.

zeta(sym(1002))
expand(zeta(sym(1002)))
ans =
zeta(1002)
ans =
(1087503...312*pi^1002)/15156647...375

### Differentiate Riemann Zeta Function

Find the third derivative of the Riemann zeta function at point x.

syms x
expr = zeta(3,x)
expr =
zeta(3, x)

Find the third derivative at x = 4 by substituting 4 for x using subs.

expr = subs(expr,x,4)
expr =
zeta(3, 4)

Evaluate expr using vpa.

expr = vpa(expr)
expr =
-0.07264084989132137196244616781177

### Plot Zeros of Riemann Zeta Function

Zeros of the Riemann Zeta function zeta(x+i*y) are found along the line x = 1/2. Plot the absolute value of the function along this line for 0<y<30 to view the first three zeros.

syms y
fplot(abs(zeta(1/2+1i*y)),[0 30])
grid on

## Input Arguments

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Input, specified as a number, vector, matrix or multidimensional array, or a symbolic number, variable, vector, matrix, multidimensional array, function or expression.

Order of derivative, specified as a nonnegative integer.

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### Riemann Zeta Function

The Riemann zeta function is defined by

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

The series converges only if the real part of z is greater than 1. The definition of the function is extended to the entire complex plane, except for a simple pole z = 1, by analytic continuation.

## Tips

• Floating point evaluation is slow for large values of n.

## Algorithms

The following exact values are implemented.

• $\zeta \left(0\right)=-\frac{1}{2}$

• $\zeta \left(1,0\right)=-\frac{\mathrm{log}\left(\pi \right)}{2}-\frac{\mathrm{log}\left(2\right)}{2}$

• $\zeta \left(\infty \right)=1$

• If $z<0$ and z is an even integer, $\zeta \left(z\right)=0.$

• If $z<0$ and z is an odd integer

$\zeta \left(z\right)=-\frac{\mathrm{bernoulli}\left(1-z\right)}{1-z}$

For $z<-1000$, zeta(z) returns an unevaluated function call. To force evaluation, use expand(zeta(z)).

• If $z>0$ and z is an even integer

$\zeta \left(z\right)=\frac{{\left(2\pi \right)}^{z}|\mathrm{bernoulli}\left(z\right)|}{2z!}$

For $z>1000$, zeta(z) returns an unevaluated function call. To force evaluation, use expand(zeta(z)).

• If $n>0$, $\zeta \left(n,\infty \right)=0.$

• If the argument does not evaluate to a listed special value, zeta returns the symbolic function call.

## Version History

Introduced before R2006a