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Finally found a fix how to stop firefox downloading each PDF to your local drive
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it isn't in right in firefox's settings -> general -> under 'files and applications' ??
'open in firefox' (what you want), vs 'always ask', 'always save' (download) or 'open in default application' (such as adobe reader). actions are configurable for many file types.
I believe even if you choose “Open in Firefox”, it will still download the PDF to the default download directory before opening it inside a Firefox tab. The behavior that OP describes above seems to prevent that downloading (and having the file around in your default download directory).