The Apple Mail client in Mavericks is reported to have some compatibility problems with Gmail. In some cases, messages are being delivered to the Gmail archive folder and are not showing up in the Inbox. In other instances, the unread / new message count indicator is incorrect. The following steps may help resolve these problems.
- Quit Mail if it is running.
- In Finder, chose Go > Go to folder.
- In the dialog that appears enter “/Library/Application Support/Apple” and press enter.
- In the new Finder window that appears there’s a folder called Mail. Drag this folder to the trash.
- Restart Mail.
What these steps accomplish is to delete some of the system settings relating to email. Be sure that you do not delete the Mail folder from your personal user Library. If you’re unsure, get a qualified support person to help you.
Thanks to Makur for discovering the solution to this problem (source discussion thread).