How you accomplish a zlib installation is entirely dependent on what distribution you use. Most distributions have some sort of package manager. RedHat and the like may use yum to do it, Debian and Ubuntu have apt-get, Gentoo has portage, and there are a number of different graphical package managers as well. I would talk to the system administrator of the machine you’re using, and ask them to install the zlib library for you.
It is possible to manually compile and install zlib in a user’s home directory, but it’s much easier if you get the sysadmin to use the distributions normal installation machinery.
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