OK, now this makes more sense. This means the notifications we receive on iOS are not generated by the app, instead the applications'/games' server pushing those directly onto the device.
I tried Altamail, and the developer said due to restrictions in place by Apple, e-mail app cannot notify the user if there is a new e-mail, and that we have to run a side-application on a desktop computer, and let that computer handle the notifications. This means the computer has to be on 24hrs to handle push notifications. I guess the developer will have to offer a paid service (few bucks a month) to keep a simple notification server running, honestly it should not cost them more than few cents to run this on a small server somewhere since all it has to do is just let the iOS device know there is a new e-mail, doesn't have to send more than a byte of data to do this.