The behaviour of the browser is based on the response type set in the response header.
This is called the MIME type and each type of file has one.
If you set the response type to
application/pdf
then it will instruct the browser to treat the file as a pdf file.
You can also set the file name using:
this.Response.AddHeader("Content-Disposition", "attachment; filename=" + file.Name);
Unfortunately, the server can't control how the browser decides to handle that file type. If you're using a browser with a PDF viewer add in. The browsers response could be to launch the viewer instead of simply downloading the file.
But the default Adobe PDF view should let you save the file.