how to clean nasty, dirty finger prints on LPs

A blowtorch judiciously applied should do the trick.

Sometimes the oils from fingerprints etch the vinyl, and nothing can be done to
eliminate them.

Try pure 99% isopropyl alky if you can still find it or straight 90% if not. Put a drop on the print and lightly spread it to cover with a corner of your record brush. Let it sit a few moments. Then brush it round with the brush as usual. Try again if it doesn't work the first time, then give up. The stuff's great for removing adhesives from CD cases too. Side note: in the United Kingdom isopropyl is kept behind the pharmacy counter to prevent young people from ingesting it for fun.

Windex. Just a bit on a microfiber cloth and rub the fingerprints gently. Follow up with a good rinse. This works well for me on used LPs where the fingerprints may have set in for a long time, mostly around the edges.