I meant in your inbox, not notifications on the phone.
Gmail already does filter the noise through "Categories" (Social, Updates, Forums, Promotions). I've turned them off as I'm pretty good about unsubscribing from junk and don't get a ton of email. However, they could place an alert at the top of your inbox to your "daily report" or whatever. Just as they have started to put an alert on incoming deliveries (ex. Amazon orders). You can then just dismiss it, so perhaps it's not an email so much as a "message" or something.
Gmail already does filter the noise through "Categories" (Social, Updates, Forums, Promotions). I've turned them off as I'm pretty good about unsubscribing from junk and don't get a ton of email. However, they could place an alert at the top of your inbox to your "daily report" or whatever. Just as they have started to put an alert on incoming deliveries (ex. Amazon orders). You can then just dismiss it, so perhaps it's not an email so much as a "message" or something.