DFU mode is also good to try first if Recovery Mode has not worked, (please note this is not the same as recovery mode) -
Plug your device into your computer.
Turn off the device.
Hold the Home button and Power button for 10 seconds
Release the Power Button but keep holding the Home button
Keep holding the Home button until you are alerted by iTunes saying that it has detected a device in Recovery Mode
Make sure the device screen is blank and no logos are present, device screen should remain black.
If DFU mode fails to work, here is the more tedious resolution -
Error 4013 is due to security software issues. Uninstall all security software completely, try again, if the issue persists I would either recommend another computer or calling AppleCare tech support at 1-800-APL-CARE (for US customers).
Note- this is even if your device worked fine prior to iOS 7 update with your security software. The change in the operating system on the device can cause compatibility issues with these other programs.
I hope this helps it worked for me!