My denon avr588 doesnt work with the remote controller

The remote control may have failed.

Read this procedure to possibly fix the remote:
http://remotes.com/remotes/servlet/rs?
a=Display&contents=help_deadremote&uuid=A12963989782821
They also sell replacement remotes.

Zoom in on the picture of the original remote provided on the site (or look at your own) and get the actual remote model number.
or check eBay: search for "(model of the device or the remote) remotes", check the 'include description' box. Usually cheaper than the vendors.

1. Press systems setup. Then press Enter.
2. Press the left or right arrow to select then press enter.
3. Press up, down, left or right and then make your selection and press enter.
4. Press enter to confirm selection.
5.

Press enter and 1 on the r/c together for more than 3secs

try to program it by using automatic search process. press and hold set up botton till the light stands on ,press the device to pprogram then press power simultaneously .When your device responds press enter botton to strore the code.

Logitech recommends that zone 2 is controlled using device mode on your Harmony remote. Controlling zone 1 and zone 2 simultaneously within activities may lead to the AV Receiver turning on and off unexpectedly as you switch between activities. For example, if you try to turn on zone 2 within an activity, it may turn on or off zone 1 and vice versa. Controlling zone 2 in device mode will prevent this from happening.

You may reset the unit to factory setting. Hold buttons "Speaker" and "SpeakerB"and turn on power switch. Release buttons after 5 seconds. You will loose all your setting and need to do configuration again.

there is a easier way to check your remote.
Open your radio on AM frequencies, you don't need to be tuned just noise is perfect, then bring your remote near the antenna and press some button. You will ear the frequency in your radio. If it works this way but not on the TV make sure to press the TV button before using it (if the remote is universal).

you should be able to disconnect the analog audio cables from the cable box and Denon, with the HDMI connected, press the input mode button on the front left of the reciever to select auto and that should fix the issue

The 1609 only has video on the HDMI. No audio comes through the HDMI cables. Hook up an optical cable between your ps3 and the receiver and you should be good to go.