Use three argument open, http://modernperlbooks.com/mt/2010/04/three-arg-open-migrating-to-modern-perl.html, use $! when calling die to tell you why the open failed.

Doesn't work by the way when I run the following code

#!c:/xampp/perl/bin/perl.exe
use warnings;
use CGI;
use Time::HiRes qw(gettimeofday);
my $query = CGI->new;
my $dir = "C:/Users/41786/OneDrive/Documents/recordz1/";
print $query->header; 

test();
 
sub test {
    my $counter = 10000; #Should be ge from db;
    my $string = "";

    open (FILE, "<$dir/test2.html") or die "cannot open file $dir/test
+2.html";
    my $content = "";
    my  $nb_page = round ($counter / 40, 1);#Number of objects display
+ed per page.
    for (my  $i = '0'; $i < $nb_page+ 1;$i++) {
        my $n2 = $i %= 10;
        $min_index += 40;    
        $string .= "<a href=\"/cgi-bin/recordz.cgi?lang=FR&amp;session
+=1&page=historique&amp;article=1amp;min_index=$min_index&amp;max_inde
+x=40\" ><-$n2-></a>&#160;&nbsp;";        
        $nb_page--;
    }
    while (<FILE>) {    
        s/\$ARTICLE{'index'}/$string/g;
        $content .= $_;    
    }
    print $content;
    close (FILE);
}

sub round {
    my $n = shift || '';
    my $r = sprintf("%.0f", $n);    
    return $r;
}
[download]

The result I'm getting in my web page is the following : <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5-> <-6-> <-7-> <-8-> <-9-> <-0-> <-1-> <-2-> <-3-> <-4-> <-5->

How does that not work?

We don't know what you expect and what you think is wrong with the output.

I assume that you pasted the rendered HTML output, but you may want to reduce your program to the bare minimum. Is the CGI part necessary for testing your program?

Update: Maybe you want to go line by line over this loop and explain what you are doing there and why:

    for (my  $i = '0'; $i < $nb_page+ 1;$i++) {
        my $n2 = $i %= 10;
        $min_index += 40;    
        $string .= "<a href=\"/cgi-bin/recordz.cgi?lang=FR&amp;session
+=1&page=historique&amp;article=1amp;min_index=$min_index&amp;max_inde
+x=40\" ><-$n2-></a>&#160;&nbsp;";        
        $nb_page--;
    }
[download]

I suggest you use a Perl-style loop instead of a C-style loop and use

for my $i (0..$nb_page) {
    ...
}
[download]

Also, why do you use my $i = '0'; instead of my $i = 0;?

Why do you decrement $nb_page at the end of the loop?

What is the $min_index variable about?

What is the $i variable supposed to hold and why do you keep it below 10?