Method for downloading a PDF from a website in Android
Android's method naming convention follows that of Java, so your method name DownloadPDF() will probably be better written as downloadPDF() (or maybe even downloadPdf()).

Your null check for uri comes in after you have toggled your UI elements, wouldn't that mean the state is possibly inconsistent if the method really received a null argument, and then the UL elements already indicated the code is downloading something? You may want to return from your method first in this case...

public void downloadPdf(URL uri) {
    if (uri == null) {
        return;
    }
    // ...
}


DownloadManager.Request methods return the object itself, so you may also consider daisy-chaining the setters together:

request.setTitle(pdf + ".pdf")
    .setDescription("Website: " + pdf)
    .setMimeType("application/pdf");