Am I required to keep a paper logbook if I use your digital logbook?

FAA FAR 61.51 states that each person must document and record certain items in a manner "acceptable to the Administrator."  The regulation does not define what is acceptable, but in most cases, digital logbooks will be accepted if the logbook records the necessary information. 

Also, the FAA has provided Advisory Circular (AC) No. 120-78 to help guide users in the user of electronic signatures and record keeping.  The AC is not mandatory and does not constitute regulation.