If you're having problems with notifications, you can run the JIRA Notification Helper to help you troubleshoot what's wrong. For that please refer the URL below.
https://confluence.atlassian.com/display/JIRACLOUD/JIRA+Admin+Helper#JIRAAdminHelper-NotificationHelper
If you want to take a look on the application logs, you should open a request on http://support.atlassian.com to request the log files.
Hope it helps!
Cheers
L.F
As Luciano said, you have to contact Support. Also, email logs are not enabled by default, so we may need to enabled then and replicate the problem to have data.
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