my iphone 6s suddenly went black. does not turn on at all. when i press home and side buttons together apple logo comes up for ten seconds and disapears by itself. have tried to restore and update thru itunes but code 4013 appeared

Hi there, see this instructions

See also this link

If you see error 9, 4005, 4013, or 4014 when you restore your iPhone, iPad, or iPod touch - Apple Support

If you see error 9, 4005, 4013, or 4014 when you restore your iPhone, iPad, or iPod touch

When you try to update or restore your iPhone, iPad, or iPod touch, you might see one of these messages on your computer.

Try to update your device again

If your device disconnects during the update or restore, or if your computer couldn't tell your device to restore, you might see one of these messages: 

• The iPhone [device name] could not be restored. An unknown error occurred (9).
• The iPhone [device name] could not be restored. An unknown error occurred (4005).
• The iPhone [device name] could not be restored. An unknown error occurred (4013).
• The iPhone [device name] could not be restored. An unknown error occurred (4014).

Try these steps:

1. Make sure that your Mac has the latest version of macOS or the latest version of iTunes.
2. Force your device to restart:

• • iPad models without a Home button: Press and quickly release the volume button closest to the top button. Press and quickly release the volume button farthest from the top button. Press and hold the top button until the device restarts. If you're not sure, see which buttons you need to press to restart your iPad.
  • iPhone 8 or later: Press and quickly release the volume up button. Press and quickly release the volume down button. Press and hold the side button until you see the Apple logo.
  • iPhone 7, iPhone 7 Plus, and iPod touch (7th generation): Press and hold both the side (or top) button and the volume down button until you see the Apple logo. 
  • iPad models with a Home button, iPhone 6s or earlier, and iPod touch (6th generation) or earlier: Press and hold both the side (or top) button and the Home button until you see the Apple logo.

1. Connect your device to your computer using a USB cable. Locate your device on your computer.
2. When you see the option to update or restore, click Update — not Restore — to reinstall the software and keep your personal data.

If you still need help, try these steps

If you can update your device but it starts up in recovery mode, follow the steps above again. On step 4, choose Restore instead of Update. After the restore completes, you need to set up your device from a backup.

If you still see an error during the update:

1. Check for updates to your Mac or PC. If you restart your computer for an update, check for updates again after you restart.
2. Try to restore your device using another USB cable.
3. Try to restore your device on another computer.

If you continue to see error 9, 4005, 4013, or 4014 when you try to restore your iPhone, iPad, or iPod touch, contact us.