I found out something that works, but i can't explain why as it seems weird.

If the path to the script delivering the PDF file is e.g. domain.com/reader/delivery.php where delivery.php reads the file and echoes it to show it in the browser, just add the filename you want to appear in the download dialog like this:

domain.com/reader/delivery.php/filename.pdf

Tested in Chrome, Firefox and Edge.