correct PHP headers for pdf file download

Example 2 on w3schools shows what you are trying to achieve.

<?php
header("Content-type:application/pdf");

// It will be called downloaded.pdf
header("Content-Disposition:attachment;filename='downloaded.pdf'");

// The PDF source is in original.pdf
readfile("original.pdf");
?>

Also remember that,

It is important to notice that header() must be called before any actual output is sent (In PHP 4 and later, you can use output buffering to solve this problem)

$name = 'file.pdf';
//file_get_contents is standard function
$content = file_get_contents($name);
header('Content-Type: application/pdf');
header('Content-Length: '.strlen( $content ));
header('Content-disposition: inline; filename="' . $name . '"');
header('Cache-Control: public, must-revalidate, max-age=0');
header('Pragma: public');
header('Expires: Sat, 26 Jul 1997 05:00:00 GMT');
header('Last-Modified: '.gmdate('D, d M Y H:i:s').' GMT');
echo $content;