How do I deal with "unauthenticated sources" errors in the Software Center?

Open the Terminal and enter the commands below one by one:

sudo apt-get clean
cd /var/lib/apt
sudo mv lists lists.old
sudo mkdir -p lists/partial
sudo apt-get clean
sudo apt-get update

Update: This might not help in some later versions, which might require alternative solutions like the one in the other answer, or this one that seems to do the job in some cases under Ubuntu 16.04:

sudo apt-get update
sudo apt-get dist-upgrade

This is a common problem with Launchpad PPAs. Common enough that I've copied these commands to use when I reinstall or add new Launchpad PPAs.

sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys  

Then,

sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys YOURKEYNUMBERHERE  

Substitute YOURKEYNUMBERHERE with the problem keys, like this:

sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys B725097B3ACC3965  

And

sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys 40976EAF437D05B5  

The first line is almost the same, minus the keys. It might even be optional, I honestly don't know. All I am sure of is that this method always fixes the problem immediately.

If anyone finds the original answer that I copied these from let me know and I'll link to it and give them the attribution.

**EDIT It's almost certainly karthik87's answer here or here I got this from. This can probably be closed as a duplicate.

Either way, don't forget to do a

sudo apt-get update  

after, to fix everything.

Note: Running apt-get update first should give you an error message mentioning which key is missing and an ID to be used with the apt-key command.

Maybe you could try changing to Main Server in Software Sources, where it says Download from.