16.04 Upgrade: Failed to fetch Empty files can't be valid archives

Go to "Ubuntu Software Center->Edit->Software Source"
select from "Download From" and choose "Other..." and click "Select Best Server". It will take some time... And it comes up with best server for you. Then chose that server and type command below again,

sudo apt-get update

I was getting this for the past few months after upgrading, but just running:

apt-get clean

fixed it. No need for changing the source servers as others have suggested.

apt-get clean

worked for me in conjunction with:

apt-key update

and following the steps found in this answer:

Cannot solve GPG error

Solved using this:

sudo cd /etc/apt
sudo mv sources.list sources.list.ok

sudo echo <<EOF
deb mirror://mirrors.ubuntu.com/mirrors.txt xenial main restricted universe multiverse
deb mirror://mirrors.ubuntu.com/mirrors.txt xenial-updates main restricted universe multiverse
deb mirror://mirrors.ubuntu.com/mirrors.txt xenial-backports main restricted universe multiverse
deb mirror://mirrors.ubuntu.com/mirrors.txt xenial-security main restricted universe multiverse
EOF > sources.list

sudo apt-get clean
sudo apt-get update

After update and upgrade, you can revert your sources.list to original !!