I have a KodakEasy Z1275 Zoom Digital Camera

The problem is with the battery of the camera its either the battery is depleted or the it doesn't have enough power to turn the camera on or maybe the memory card is also corrupted.

try to take out the battery and the memory card
wipe the battery compartment with a lint free cloth and try to put back the battery without the memory card and try to turn it on if the problem persist try a different lithium battery

you definitely have a battery issue , you either not using the right battery or your batteries are not fully charged.

Assuming that you have fresh or fully charged batteries installed, chances are that the problem is corroded battery contacts inside the camera. Remove the batteries and clean the inside contacts with a fine wire brush, steel wool, or fine sandpaper. Remove any residue that's in the camera from the cleaning, then reinstall your batteries. Hopefully this will cure the problem and get the camera back up and working the way it should.

If that doesn't do the trick, you might want to try your luck with the Kodak Troubleshooting guide at:

http://www.kodak.com/eknec/PageQuerier.jhtml?pq-path=10&pq-locale=en_US