In Microsoft Outlook for iOS devices, you can disable receiving notifications for e-mail while continuing receiving notifications for your calendar and calendar invites.
FIrst, open Outlook on your device. Locate your Inbox in Outlook and click it to bring up a complete list of folders. In this view, in the bottom left corner, you will see a gear symbol that is your access to settings for Outlook. Click on the gear.
While scrolling down, you will first arrive at a subsection of settings for "Mail". The first option is "Notifications". Here, you can change your notification preferences, or change it to "None" to receive zero e-mail notifications.
Next, click the back arrow to come back to the main Outlook settings screen. Continue to scroll down past "Contacts" until you arrive at "Calendar". Here, you can again choose the "Notifications" section to set your preferences to customize calendar messages.