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Thanks for the feedback.
This is Tim with Microsoft Office 365 SharePoint Support.
The "open" or "save" when trying to open a hyper linked PDF file is by design. This is because PDF files were deemed a security risk with the way that permissive file handling works. In an up coming service update of SharePoint Online, PDF files will have the ability to automatically be opened in the client application when opened via a hyper link from your public-facing site.
I will continue to monitor this thread if you have any additional questions.
Microsoft Office 365 SharePoint Support