That's weird. What happens when you try to open Outlook on the second computer?

Do you have your .pst files on a network drive?

Since it's looking for your .pst file on the Z: drive i thought that maybe you put your pst file there. Where is your Z: drive? Is that another partition on your Hard Drive or another Hard drive all together?

If you are using POP3 to access your email then they will not be in sync on the 2 different computers.
You can use IMAP instead.

Is it a GMAIL account you are accessing?

The default location for outlook .pst files is usually something like :
"C:\Users\your user name here\AppData\Local\Microsoft\Outlook"