Multiple select and AJAX submit

Multiple select and AJAX submit

I'm trying to submit a form with ajax.

HTML code (this is a jquery multiselect, I can select from 1 to 3 options):
<form action="" method="post" id="formzone">
<select id="zone" name="zone" multiple="multiple" size="5">
<option value="1">OPTION 1</option>
<option value="2">OPTION 2</option>
<option value="3">OPTION 3</option>
<option value="4">OPTION 4</option>
</select>
<input type="submit" value="Show Results" />
</form>
<div id="filterzone"></div>


js code:
$('#formzone').submit(function() {
var results = $(this).serialize();
$("#filterzone").text(results);
return false; //these last two lines wont be there when I add ajax


This works, I think: if I select options 1, 2 and 4, the text in #filterzone becomes:

zone=1&zone=2&zone=4

I'm now trying to add AJAX and display the response in that div.
The response comes from this simple page (just to see if it works...):
filterzone.php:
<?php
print_r ($_POST);
?>

Ajax code:

             $.ajax({
               type: "POST",
               data: results,
               url: "filterzone.php",
               error: function() {
                  $("#filterzone").text("Failed to submit");
               },
               success: function(r) {
                  $("#filterzone").text(r);
            }
        });
            
         });


It says "Failed to submit" for a moment, then refreshes the page.
What am I doing wrong? I looked around for some tutorials, but they're always about a login form (so with a fixed number of inputs, I could have thousands of options instead, although i submit just three of them).

Thank you