When I turn the camera on, it buzzes four or five times (a sound I have never heard it make before) then displays "Please turn the camera off and then on again". When I turn it off, it buzzes again two or three times, the screen goes blank, but the lens does not retract. The battery is charged, and the problem is the same whether in picture or display mode, whether plugged into the computer or not.

Fixing a Lens Error on a Digital Camera

This has to be THE most common failure mode for a digital camera. Some common error messages that might show up on the LCD's of cameras with this problem include “E18 lens error”, or “lens error, restart camera”. Some cameras might show nothing at all, but merely make a beeping noise as the lens goes out, then in, then the camera shuts off. Sometimes the lens won't even move.
The problem is actually quite common throughout all camera brands. Usually it's sand or grit interfering with the lens extension mechanism. Or the camera's been dropped with the lens extended. Or the camera has been powered on, but the lens had been blocked preventing its extension. Or the battery ran down with the lens extended. Believe it or not, one BIG contributor to lens errors is using a camera case. Sand, gunk, case fibers, etc... accumulate at the bottom of the case. These materials love to cling to the camera by electrostatic build-up from the camera rubbing against the side of the case (especially those cases with soft fibrous intreriors). Once these materials work their way into the lens mechanism, that's all she wrote. I have many Canon's, and NEVER use a case for this very reason.
A camera owner that suffers this problem may have no recourse for having the camera repaired. Many camera makers will not honor repairing this problem under warranty as they claim it is due to impact damage to the camera, or sand or debris getting into the lens gearing mechanism (neither of which is covered under warranty). The quoted repair cost is usually close to or more than what the camera is actually worth.
Fortunately, about half the cameras that suffer this failure can easily be fixed by one of the following methods. None of these methods involve opening the camera, although some have potential to cause other damage to the camera if excessively done. If the camera is still under warranty, before trying any of these, please please first contact your camera's maker to see if they'll cover the repair, or to determine how much they'll charge for the repair. Who knows, you might get lucky. But if they quote you a number that's higher than the value of your camera, you may want to consider the following methods.
The methods are listed in the order of risk of damaging your camera. Thus make sure you try them in the listed order. And remember, these fixes (especially #6 and 7) should only be considered for a camera that's out of warranty, who's cost of repair would be excessive, and would otherwise be considered for disposal if unrepaired:
Fix #1: Remove the batteries from the camera, wait a few minutes. Put a fresh set of batteries back in (preferably rechargeable NiMH 2500mah or better) and turn the camera on. If that didn't work, try pressing and holding the Function or OK button while turning the camera on.
Fix #2: Remove the batteries, then remove the memory card. Then install new batteries, and turn on the camera. If you get an Error E30, it means you don't have a memory card installed, so turn it off, slip in the memory card and turn it on one last time.
Fix #3: Insert the cameras Audio/Video (AV) cable, and turn the camera on. Inserting this cable ensures that the camera's LCD screen remains off during the start process. Thus extra battery power is available to the camera's lens motor during startup. This extra power can be useful in overcoming grit or sand particals that may be jamming the lens. If the AV cable doesn't fix the lens error by itself, consider keeping this cable installed while trying fixes 4, 5, and 7 as a means to provide extra help to these fixes. But note that I DON'T recommend keeping the cable installed during Fix 6 as you may damage the AV port while tapping the camera. Reinsert the cable only AFTER tapping the camera.
Fix #4: Place the camera flat on its back on a table, pointed at the ceiling. Press and hold the shutter button down, and at the same time press the power-on button. The idea is that the camera will try to autofocus while the lens is extending, hopefully seating the lens barrel guide pins in their slots.
Fix #5: Blow compressed air in the gaps around the lens barrels with the idea of blowing out any sand or grit that may be in there jamming the lens. Other variations include blowing with a hair dryer in “no heat” setting, or sucking the gaps with a vacuum (careful with this one).
Now we're entering into the realm of potentially damaging your camera in conducting the fix. There is definitely some risk here, so take care when conducting the following two fixes.
Fix #6: Repeatedly tap the padded/rubber usb cover on a hard surface with the intent of dislodging any particles that may be jamming the lens. Other variations include hitting a side of the camera against the palm of your hand. A lot of people have reported success with this method. HOWEVER, there is also some potential for damaging or dislodging internal components with this method, such as unseating ribbon cables, or cracking LCD screens.
Fix #7: Try forcing the lens. More people have reported success with this method than with any of the other methods. HOWEVER, there's obviously some potential for damaging your camera by using this method. Variations include gently pulling, rotating, and/or twisting the lens barrel while hitting the power button. Attempt to gently straighten or align the barrel if it's crooked or twisted. Another variation includes looking for uneven gaps around the lens barrel, and then pushing on the side of the lens barrel that has the largest gap (note pushing the lens barrel all the way in is NOT recommended as it may become stuck there). While doing any of the above, listen for a click that indicates that the lens barrel guide pins may have reseated in their guide slots. If you hear this click, immediately stop and try the camera.