There are probably better ways to do this, but I thought of this off the top of my head.
It uses two files: one with link to the .pdf file, the other a counter.
1) 'download.php'
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>Download Script</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
<link href="css/style.css" rel="stylesheet" type="text/css" />
</head>
<body>
<?php
// Change THIS to your file name!
$file = 'sample.pdf';
$phpcounter = 'counter.php';
$now = date('F dS, Y (g:i a)');
if(isset($_GET['action']) && $_GET['action'] == 'download'){
$handle = fopen($phpcounter, 'r');
$oldcontents = fread($handle, filesize($phpcounter));
fclose($handle);
$handle = fopen($phpcounter, 'w');
$newcontents = str_replace('?>', '', $oldcontents);
$newcontents .= '$downloads[] = \'' . $now . '\';' . "\n";
$newcontents .= '?>';
$results = fwrite($handle, $newcontents);
fclose($handle);
?>
<iframe name="box" src="<?php echo $file; ?>" height="0" width="0" frameborder="0"></iframe>
<p style="margin:150px;font-family:arial;text-align:center;font-size:12px;">
[<a href="javascript:;" onClick="self.close();return false">Close Window</a>]</p>
<?php
}
else {
include_once $phpcounter;
$count = (count($downloads) - 1);
?>
<p><a href="<?php echo $_SERVER['PHP_SELF'] . '?action=download'; ?>" target="box">Download file now!</a>
[<small>Downloaded <?php echo $count; ?></small> times]</p>
<?php } ?>
</body>
</html>
2) 'counter.php'
<?php
$downloads[] = 'NULL';
?>
The idea is that if the page is just visited, the counter ('counter.php') will be included and the contents read as an array. The array items will be counted and echoed onto the page.
However, if the link to download the file is clickec, a new window opens, the download begins in a hidden iframe, and the counter ('counter.php') will be updated with a timestamp, so you will know how may downloads AND when they were downloaded.
I hope this gives you a starting point.