# [libmath-prime-util-perl] 03/20: Slight doc change for moebius/phi/lambda/M

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Author: Dana Jacobsen <d...@acm.org>
Date:   Wed Feb 27 09:42:14 2013 -0800

Slight doc change for moebius/phi/lambda/M
---
lib/Math/Prime/Util.pm | 26 +++++++++++++-------------
t/80-pp.t              |  6 +++++-
2 files changed, 18 insertions(+), 14 deletions(-)

diff --git a/lib/Math/Prime/Util.pm b/lib/Math/Prime/Util.pm
index f63495f..7e76610 100644
--- a/lib/Math/Prime/Util.pm
+++ b/lib/Math/Prime/Util.pm
@@ -2414,8 +2414,8 @@ primality tests.
say "\$n is square free" if moebius(\$n) != 0;
\$sum += moebius(\$_) for (1..200); say "Mertens(200) = \$sum";

-Returns the Möbius function (also called the Moebius, Mobius, or MoebiusMu
-function) for a non-negative integer input.  This function is 1 if
+Returns μ(n), the Möbius function (also called the Moebius, Mobius, or
+MoebiusMu function) for a non-negative integer input.  This function is 1 if
C<n = 1>, 0 if C<n> is not square free (i.e. C<n> has a repeated factor),
and C<-1^t> if C<n> is a product of C<t> distinct primes.  This is an
important function in prime number theory.  Like SAGE, we define
@@ -2432,7 +2432,7 @@ which is a segmented version of Lioen and van de Lune
(1994) algorithm 3.2.

say "Mertens(10M) = ", mertens(10_000_000);   # = 1037

-Returns the Mertens function of the positive non-zero integer input.  This
+Returns M(n), the Mertens function for a non-negative integer input.  This
function is defined as C<sum(moebius(1..n))>.  This is a much more efficient
solution for larger inputs.  For example, computing Mertens(100M) takes:

@@ -2457,10 +2457,10 @@ algorithms can lead to a faster solution.

say "The Euler totient of \$n is ", euler_phi(\$n);

-Returns the Euler totient function (also called Euler's phi or phi function)
-for an integer value.  This is an arithmetic function that counts the number
-of positive integers less than or equal to C<n> that are relatively prime to
-C<n>.  Given the definition used, C<euler_phi> will return 0 for all
+Returns φ(n), the Euler totient function (also called Euler's phi or phi
+function) for an integer value.  This is an arithmetic function that counts
+the number of positive integers less than or equal to C<n> that are relatively
+prime to C<n>.  Given the definition used, C<euler_phi> will return 0 for all
C<n E<lt> 1>.  This follows the logic used by SAGE.  Mathematic/WolframAlpha
also returns 0 for input 0, but returns C<euler_phi(-n)> for C<n E<lt> 0>.

@@ -2486,12 +2486,12 @@ the Dedikind psi function, where C<psi(n) = J(2,n) /
J(1,n)>.

say "exp(lambda(\$_)) = ", exp_mangoldt(\$_) for 1 .. 100;

-The Mangoldt function Λ(n) (also known as von Mangoldt's function) is equal
-to log p if n is prime or a power of a prime, and 0 otherwise.  We return
-the exponential so all results are integers.  Hence the return value
-for C<exp_mangoldt> is:
-   C<p> if C<n = p^m> for some prime C<p> and integer C<m E<gt>= 1>
-   1 otherwise.
+Returns Λ(n), the Mangoldt function (also known as von Mangoldt's function) for
+an integer value.  It is equal to log p if n is prime or a power of a prime,
+and 0 otherwise.  We return the exponential so all results are integers.
+ Hence the return value for C<exp_mangoldt> is:
+   C<p>   if C<n = p^m> for some prime C<p> and integer C<m E<gt>= 1>
+   1   otherwise.

diff --git a/t/80-pp.t b/t/80-pp.t
index 1ec86d5..06f7c92 100644
--- a/t/80-pp.t
+++ b/t/80-pp.t
@@ -226,7 +226,7 @@ plan tests => 1 +
1 + 1 +    # factor
8 + 4*3 +  # factoring subs
10 +       # AKS
-              0;
+              1;

use Math::Prime::Util qw/primes prime_count_approx prime_count_lower/;
use Math::BigInt try => 'GMP';
@@ -253,6 +253,8 @@ require_ok 'Math::Prime::Util::PP';

###############################################################################

+\$_ = 'this should not change';
+
{
my %small_primes = map { \$_ => 1 } @small_primes;
my @isprime = map { is_prime(\$_) } (0 .. 1086);
@@ -471,6 +473,8 @@ SKIP: {
is( is_aks_prime(74513), 0, "AKS: 74513 is composite (failed anr test)" );
}

+is( \$_, 'this should not change', "Nobody clobbered \\$_" );
+
###############################################################################

sub parse_range {

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