See New Gmail Messages in the iOS Notification Center
To have new emails coming into your Gmail account listed and previewed in your iPhone or iPad’s Notification Center:
Additional iOS Notification Tweaks for Gmail
After turning on the notifications, you can adjust a few more settings to get as many—or as few—alerts as you want. Here’s what you can control.
See Notifications on the Lock Screen and While Using Your Phone
To get additional alerts outside of the Notification Center, tap the Lock Screen and Banners options under the Alerts heading. The first setting lets you see incoming Gmail messages without unlocking your iPhone. The second displays a window on your phone while you’re using another app. You can also adjust how the banners behave. Tap Banner Style to access two options: Temporary and Permanent. Temporary means that the banners disappear after a while, and Permanent means that they stay on the screen until you tap or swipe them away.
Adjust Sounds and Badges for Gmail Notifications
The Sounds and Badges switches control additional alerts for your notifications in Gmail. Sounds plays a tone to alert you to the notification. Badges puts a number on the app icon on your home screen.
Notifications Settings in the Gmail App
The Gmail app has its own settings for the messages you’ll receive alerts for. Here’s how to adjust them. You have three options. All New Mail sends you everything that arrives. High Priority Only alerts you of messages that Gmail flags as important. None turns off notifications.