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when I get a new notification I get a badge notification on my icon. However, is there a way to keep the number of unread emails on the badge Icon.
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That will depend on the App you use for EMail.
The native GMail for example works that way but others I've tried do it.
It's likely due to how the App itself tracks Notifications since most weren't designed to be used in this way.
However, apparently Android 4.3 will have a built-in setting for "Notifiers" which theoretically could allow for it to work the way you described regardless of the EMail client you use.
In the meantime I'd recommend you contact the Dev of the EMail App you want to act this way (informing them of the discrepancies between theirs & GMail's) to see if they can implement it.
Sent from my Nexus 4 using AC Forums mobile app