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<h1 class="chapter"> 32. Number Theory </h1>


<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC129">32.1 Definitions for Number Theory</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">  
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<h2 class="section"> 32.1 Definitions for Number Theory </h2>

<dl>
<dt><u>Function:</u> <b>bern</b><i> (<var>n</var>)</i>
<a name="IDX1046"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Bernoulli number for integer <var>n</var>.
Bernoulli numbers equal to zero are suppressed if <code>zerobern</code> is <code>false</code>.
</p>
<p>See also <code>burn</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) zerobern: true$
(%i2) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                  1  1       1      1        1
(%o2)       [1, - -, -, 0, - --, 0, --, 0, - --]
                  2  6       30     42       30
(%i3) zerobern: false$
(%i4) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
            1  1    1   5     691   7    3617  43867
(%o4) [1, - -, -, - --, --, - ----, -, - ----, -----]
            2  6    30  66    2730  6    510    798
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>bernpoly</b><i> (<var>x</var>, <var>n</var>)</i>
<a name="IDX1047"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Bernoulli polynomial in the
variable <var>x</var>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>bfzeta</b><i> (<var>s</var>, <var>n</var>)</i>
<a name="IDX1048"></a>
</dt>
<dd><p>Returns the Riemann zeta function for the argument <var>s</var>.
The return value is a big float (bfloat);
<var>n</var> is the number of digits in the return value.
</p>
<p><code>load (&quot;bffac&quot;)</code> loads this function.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>bfhzeta</b><i> (<var>s</var>, <var>h</var>, <var>n</var>)</i>
<a name="IDX1049"></a>
</dt>
<dd><p>Returns the Hurwitz zeta function for the arguments <var>s</var> and <var>h</var>.
The return value is a big float (bfloat);
<var>n</var> is the number of digits in the return value.
</p>
<p>The Hurwitz zeta function is defined as
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">sum ((k+h)^-s, k, 0, inf)
</pre></td></tr></table>
<p><code>load (&quot;bffac&quot;)</code> loads this function.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>binomial</b><i> (<var>x</var>, <var>y</var>)</i>
<a name="IDX1050"></a>
</dt>
<dd><p>The binomial coefficient <code><var>x</var>!/(<var>y</var>! (<var>x</var> - <var>y</var>)!)</code>.
If <var>x</var> and <var>y</var> are integers, then the numerical value of the binomial
coefficient is computed.
If <var>y</var>, or <var>x - y</var>, is an integer,
the binomial coefficient is expressed as a polynomial.
</p>
<p>Examples:
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) binomial (11, 7);
(%o1)                          330
(%i2) 11! / 7! / (11 - 7)!;
(%o2)                          330
(%i3) binomial (x, 7);
        (x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
(%o3)   -------------------------------------------------
                              5040
(%i4) binomial (x + 7, x);
      (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
(%o4) -------------------------------------------------------
                               5040
(%i5) binomial (11, y);
(%o5)                    binomial(11, y)
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>burn</b><i> (<var>n</var>)</i>
<a name="IDX1051"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Bernoulli number for integer <var>n</var>.
<code>burn</code> may be more efficient than <code>bern</code> for large, isolated <var>n</var>
(perhaps <var>n</var> greater than 105 or so), as <code>bern</code> computes all the Bernoulli numbers up to index <var>n</var> before returning.
</p>

<p><code>burn</code> exploits the observation that (rational) Bernoulli numbers can be
approximated by (transcendental) zetas with tolerable efficiency.
</p>
<p><code>load (&quot;bffac&quot;)</code> loads this function.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>cf</b><i> (<var>expr</var>)</i>
<a name="IDX1052"></a>
</dt>
<dd><p>Converts <var>expr</var> into a continued fraction.
<var>expr</var> is an expression
comprising continued fractions and square roots of integers.
Operands in the expression may be combined with arithmetic operators.
Aside from continued fractions and square roots,
factors in the expression must be integer or rational numbers.
Maxima does not know about operations on continued fractions outside of <code>cf</code>.
</p>
<p><code>cf</code> evaluates its arguments after binding <code>listarith</code> to <code>false</code>.
<code>cf</code> returns a continued fraction, represented as a list.
</p>
<p>A continued fraction <code>a + 1/(b + 1/(c + ...))</code>
is represented by the list <code>[a, b, c, ...]</code>.
The list elements <code>a</code>, <code>b</code>, <code>c</code>, ... must evaluate to integers.
<var>expr</var> may also contain <code>sqrt (n)</code> where <code>n</code> is an integer.
In this case <code>cf</code> will give as many
terms of the continued fraction as the value of the variable
<code>cflength</code> times the period.
</p>
<p>A continued fraction can be evaluated to a number
by evaluating the arithmetic representation
returned by <code>cfdisrep</code>.
See also <code>cfexpand</code> for another way to evaluate a continued fraction.
</p>
<p>See also <code>cfdisrep</code>, <code>cfexpand</code>, and <code>cflength</code>.
</p>
<p>Examples:
</p>
<ul>
<li>
<var>expr</var> is an expression comprising continued fractions and square roots of integers.

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]);
(%o1)               [59, 17, 2, 1, 1, 1, 27]
(%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13));
(%o2)        [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]
</pre></td></tr></table>
</li><li>
<code>cflength</code> controls how many periods of the continued fraction
are computed for algebraic, irrational numbers.

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) cflength: 1$
(%i2) cf ((1 + sqrt(5))/2);
(%o2)                    [1, 1, 1, 1, 2]
(%i3) cflength: 2$
(%i4) cf ((1 + sqrt(5))/2);
(%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
(%i5) cflength: 3$
(%i6) cf ((1 + sqrt(5))/2);
(%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
</pre></td></tr></table>
</li><li>
A continued fraction can be evaluated by evaluating the arithmetic representation
returned by <code>cfdisrep</code>.

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) cflength: 3$
(%i2) cfdisrep (cf (sqrt (3)))$
(%i3) ev (%, numer);
(%o3)                   1.731707317073171
</pre></td></tr></table>
</li><li>
Maxima does not know about operations on continued fractions outside of <code>cf</code>.

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) cf ([1,1,1,1,1,2] * 3);
(%o1)                     [4, 1, 5, 2]
(%i2) cf ([1,1,1,1,1,2]) * 3;
(%o2)                  [3, 3, 3, 3, 3, 6]
</pre></td></tr></table>
</li></ul>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>cfdisrep</b><i> (<var>list</var>)</i>
<a name="IDX1053"></a>
</dt>
<dd><p>Constructs and returns an ordinary arithmetic expression
of the form <code>a + 1/(b + 1/(c + ...))</code>
from the list representation of a continued fraction <code>[a, b, c, ...]</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1)                     [1, 1, 1, 2]
(%i2) cfdisrep (%);
                                  1
(%o2)                     1 + ---------
                                    1
                              1 + -----
                                      1
                                  1 + -
                                      2
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>cfexpand</b><i> (<var>x</var>)</i>
<a name="IDX1054"></a>
</dt>
<dd><p>Returns a matrix of the numerators and denominators of the
last (column 1) and next-to-last (column 2) convergents of the continued fraction <var>x</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) cf (rat (ev (%pi, numer)));

`rat' replaced 3.141592653589793 by 103993//33102 = 3.141592653011902
(%o1)                  [3, 7, 15, 1, 292]
(%i2) cfexpand (%); 
                         [ 103993  355 ]
(%o2)                    [             ]
                         [ 33102   113 ]
(%i3) %[1,1]/%[2,1], numer;
(%o3)                   3.141592653011902
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>cflength</b>
<a name="IDX1055"></a>
</dt>
<dd><p>Default value: 1
</p>
<p><code>cflength</code> controls the number of terms of the continued
fraction the function <code>cf</code> will give, as the value <code>cflength</code> times the
period.  Thus the default is to give one period.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) cflength: 1$
(%i2) cf ((1 + sqrt(5))/2);
(%o2)                    [1, 1, 1, 1, 2]
(%i3) cflength: 2$
(%i4) cf ((1 + sqrt(5))/2);
(%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
(%i5) cflength: 3$
(%i6) cf ((1 + sqrt(5))/2);
(%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>divsum</b><i> (<var>n</var>, <var>k</var>)</i>
<a name="IDX1056"></a>
</dt>
<dt><u>Function:</u> <b>divsum</b><i> (<var>n</var>)</i>
<a name="IDX1057"></a>
</dt>
<dd><p><code>divsum (<var>n</var>, <var>k</var>)</code> returns the sum of the divisors of <var>n</var>
raised to the <var>k</var>'th power.
</p>
<p><code>divsum (<var>n</var>)</code> returns the sum of the divisors of <var>n</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) divsum (12);
(%o1)                          28
(%i2) 1 + 2 + 3 + 4 + 6 + 12;
(%o2)                          28
(%i3) divsum (12, 2);
(%o3)                          210
(%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
(%o4)                          210
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>euler</b><i> (<var>n</var>)</i>
<a name="IDX1058"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Euler number for nonnegative integer <var>n</var>.
</p>
<p>For the Euler-Mascheroni constant, see <code>%gamma</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) map (euler, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
(%o1)    [1, 0, - 1, 0, 5, 0, - 61, 0, 1385, 0, - 50521]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Constant:</u> <b>%gamma</b>
<a name="IDX1059"></a>
</dt>
<dd><p>The Euler-Mascheroni constant, 0.5772156649015329 ....
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>factorial</b><i> (<var>x</var>)</i>
<a name="IDX1060"></a>
</dt>
<dd><p>Represents the factorial function. Maxima treats <code>factorial (<var>x</var>)</code> the same as <code><var>x</var>!</code>.
See <code>!</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fib</b><i> (<var>n</var>)</i>
<a name="IDX1061"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Fibonacci number.
<code>fib(0)</code> equal to 0 and <code>fib(1)</code> equal to 1,
and
<code>fib (-<var>n</var>)</code> equal to <code>(-1)^(<var>n</var> + 1) * fib(<var>n</var>)</code>.
</p>
<p>After calling <code>fib</code>,
<code>prevfib</code> is equal to <code>fib (<var>x</var> - 1)</code>,
the Fibonacci number preceding the last one computed.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) map (fib, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
(%o1)         [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>fibtophi</b><i> (<var>expr</var>)</i>
<a name="IDX1062"></a>
</dt>
<dd><p>Expresses Fibonacci numbers in terms of the constant <code>%phi</code>,
which is <code>(1 + sqrt(5))/2</code>, approximately 1.61803399.
</p>
<p>By default, Maxima does not know about <code>%phi</code>.
After executing <code>tellrat (%phi^2 - %phi - 1)</code> and <code>algebraic: true</code>,
<code>ratsimp</code> can simplify some expressions containing <code>%phi</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) fibtophi (fib (n));
                           n             n
                       %phi  - (1 - %phi)
(%o1)                  -------------------
                           2 %phi - 1
(%i2) fib (n-1) + fib (n) - fib (n+1);
(%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
(%i3) ratsimp (fibtophi (%));
(%o3)                           0
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>ifactors</b><i> (<var>n</var>)</i>
<a name="IDX1063"></a>
</dt>
<dd><p>For a positive integer <var>n</var> returns the factorization of <var>n</var>. If
<code>n=p1^e1..pk^nk</code> is the decomposition of <var>n</var> into prime
factors, ifactors returns <code>[[p1, e1], ... , [pk, ek]]</code>.
</p>
<p>Factorization methods used are trial divisions by primes up to 9973,
Pollard's rho method and elliptic curve method.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) ifactors(51575319651600);
(%o1)     [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
(%i2) apply(&quot;*&quot;, map(lambda([u], u[1]^u[2]), %));
(%o2)                        51575319651600
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>inrt</b><i> (<var>x</var>, <var>n</var>)</i>
<a name="IDX1064"></a>
</dt>
<dd><p>Returns the integer <var>n</var>'th root of the absolute value of <var>x</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
(%i2) map (lambda ([a], inrt (10^a, 3)), l);
(%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>inv_mod</b><i> (<var>n</var>, <var>m</var>)</i>
<a name="IDX1065"></a>
</dt>
<dd><p>Computes the inverse of <var>n</var> modulo <var>m</var>. 
<code>inv_mod (n,m)</code> returns <code>false</code>, 
if <var>n</var> is a zero divisor modulo <var>m</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) inv_mod(3, 41);
(%o1)                           14
(%i2) ratsimp(3^-1), modulus=41;
(%o2)                           14
(%i3) inv_mod(3, 42);
(%o3)                          false
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>jacobi</b><i> (<var>p</var>, <var>q</var>)</i>
<a name="IDX1066"></a>
</dt>
<dd><p>Returns the Jacobi symbol of <var>p</var> and <var>q</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
(%i2) map (lambda ([a], jacobi (a, 9)), l);
(%o2)         [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>lcm</b><i> (<var>expr_1</var>, ..., <var>expr_n</var>)</i>
<a name="IDX1067"></a>
</dt>
<dd><p>Returns the least common multiple of its arguments.
The arguments may be general expressions as well as integers.
</p>
<p><code>load (&quot;functs&quot;)</code> loads this function.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>minfactorial</b><i> (<var>expr</var>)</i>
<a name="IDX1068"></a>
</dt>
<dd><p>Examines <var>expr</var> for occurrences of two factorials
which differ by an integer.
<code>minfactorial</code> then turns one into a polynomial times the other.
</p>

<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) n!/(n+2)!;
                               n!
(%o1)                       --------
                            (n + 2)!
(%i2) minfactorial (%);
                                1
(%o2)                    ---------------
                         (n + 1) (n + 2)
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>power_mod</b><i> (<var>a</var>, <var>n</var>, <var>m</var>)</i>
<a name="IDX1069"></a>
</dt>
<dd><p>Uses a modular algorithm to compute <code>a^n mod m</code> 
where <var>a</var> and <var>n</var> are integers and <var>m</var> is a positive integer. 
If <var>n</var> is negative, <code>inv_mod</code> is used to find the modular inverse.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) power_mod(3, 15, 5);
(%o1)                          2
(%i2) mod(3^15,5);
(%o2)                          2
(%i3) power_mod(2, -1, 5);
(%o3)                          3
(%i4) inv_mod(2,5);
(%o4)                          3
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>next_prime</b><i> (<var>n</var>)</i>
<a name="IDX1070"></a>
</dt>
<dd><p>Returns the smallest prime bigger than <var>n</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) next_prime(27);
(%o1)                       29
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>partfrac</b><i> (<var>expr</var>, <var>var</var>)</i>
<a name="IDX1071"></a>
</dt>
<dd><p>Expands the expression <var>expr</var> in partial fractions
with respect to the main variable <var>var</var>.  <code>partfrac</code> does a complete
partial fraction decomposition.  The algorithm employed is based on
the fact that the denominators of the partial fraction expansion (the
factors of the original denominator) are relatively prime.  The
numerators can be written as linear combinations of denominators, and
the expansion falls out.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
                      2       2        1
(%o1)               ----- - ----- + --------
                    x + 2   x + 1          2
                                    (x + 1)
(%i2) ratsimp (%);
                                 x
(%o2)                 - -------------------
                         3      2
                        x  + 4 x  + 5 x + 2
(%i3) partfrac (%, x);
                      2       2        1
(%o3)               ----- - ----- + --------
                    x + 2   x + 1          2
                                    (x + 1)
</pre></td></tr></table></dd></dl>

<dl>
<dt><u>Function:</u> <b>primep</b><i> (<var>n</var>)</i>
<a name="IDX1072"></a>
</dt>
<dd><p>Primality test. If <code>primep (n)</code> returns <code>false</code>, <var>n</var> is a
composite number and if it returns <code>true</code>, <var>n</var> is a prime number
with very high probability.
</p>
<p>For <var>n</var> less than 341550071728321 a deterministic version of Miller-Rabin's
test is used. If <code>primep (n)</code> returns <code>true</code>, then <var>n</var> is a
prime number.
</p>
<p>For <var>n</var> bigger than 34155071728321 <code>primep</code> uses
<code>primep_number_of_tests</code> Miller-Rabin's pseudo-primality tests
and one Lucas pseudo-primality test. The probability that <var>n</var> will
pass one Miller-Rabin test is less than 1/4. Using the default value 25 for
<code>primep_number_of_tests</code>, the probability of <var>n</var> beeing
composite is much smaller that 10^-15.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>primep_number_of_tests</b>
<a name="IDX1073"></a>
</dt>
<dd><p>Default value: 25
</p>
<p>Number of Miller-Rabin's tests used in <code>primep</code>.
</p></dd></dl>

<dl>
<dt><u>Function:</u> <b>prev_prime</b><i> (<var>n</var>)</i>
<a name="IDX1074"></a>
</dt>
<dd><p>Returns the greatest prime smaller than <var>n</var>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) prev_prime(27);
(%o1)                       23
</pre></td></tr></table></dd></dl>

<dl>
<dt><u>Function:</u> <b>qunit</b><i> (<var>n</var>)</i>
<a name="IDX1075"></a>
</dt>
<dd><p>Returns the principal unit of the real quadratic number field
<code>sqrt (<var>n</var>)</code> where <var>n</var> is an integer,
i.e., the element whose norm is unity.
This amounts to solving Pell's equation <code>a^2 - <var>n</var> b^2 = 1</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) qunit (17);
(%o1)                     sqrt(17) + 4
(%i2) expand (% * (sqrt(17) - 4));
(%o2)                           1
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>totient</b><i> (<var>n</var>)</i>
<a name="IDX1076"></a>
</dt>
<dd><p>Returns the number of integers less than or equal to <var>n</var> which
are relatively prime to <var>n</var>.
</p>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>zerobern</b>
<a name="IDX1077"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>zerobern</code> is <code>false</code>,
<code>bern</code> excludes the Bernoulli numbers which are equal to zero. 
See <code>bern</code>.
</p>
</dd></dl>

<dl>
<dt><u>Function:</u> <b>zeta</b><i> (<var>n</var>)</i>
<a name="IDX1078"></a>
</dt>
<dd><p>Returns the Riemann zeta function if <var>x</var> is a negative integer, 0, 1,
or a positive even number,
and returns a noun form <code>zeta (<var>n</var>)</code> for all other arguments,
including rational noninteger, floating point, and complex arguments.
</p>
<p>See also <code>bfzeta</code> and <code>zeta%pi</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) map (zeta, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5]);
                                     2              4
           1        1     1       %pi            %pi
(%o1) [0, ---, 0, - --, - -, inf, ----, zeta(3), ----, zeta(5)]
          120       12    2        6              90
</pre></td></tr></table>
</dd></dl>

<dl>
<dt><u>Option variable:</u> <b>zeta%pi</b>
<a name="IDX1079"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>zeta%pi</code> is <code>true</code>, <code>zeta</code> returns an expression
proportional to <code>%pi^n</code> for even integer <code>n</code>.
Otherwise, <code>zeta</code> returns a noun form <code>zeta (n)</code>
for even integer <code>n</code>.
</p>
<table><tr><td>&nbsp;</td><td><pre class="example">(%i1) zeta%pi: true$
(%i2) zeta (4);
                                 4
                              %pi
(%o2)                         ----
                               90
(%i3) zeta%pi: false$
(%i4) zeta (4);
(%o4)                        zeta(4)
</pre></td></tr></table>
</dd></dl>

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