B757-200 Captain reported fluctuating oil pressure and zero oil quantity with associated EICAS messages on the right engine after takeoff. The flight crew ran QRH procedure; followed checklists; and performed an air turn back and precautionary landing.

Narrative

During climb-out from ZZZ; passing approximately 17;000 ft. MSL; received a master caution light accompanied by a R ENG OIL PRESS EICAS (Engine Indicating and Crew Alerting System) message. A check of the EICAS showed that the right engine oil quantity was zero and the oil pressure was fluctuating... mostly above the amber caution zone. By the time I pulled out the QRH; the oil pressure indication was dipping into the amber band repeatedly. The FO (First Officer) was the pilot flying and she continued to fly while I accomplished the R ENG OIL PRESS QRH procedure. While accomplishing the QRH; the oil pressure would dip lower and actually momentarily hit the red line lower limit so the QRH directed us to proceed to the Engine Failure or Shutdown Checklist. We ran that checklist and shut down the right engine. Oil quantity indicated zero but there was some oil pressure indicating when we shut it down. As I was running the checklists; we had already requested a return to ZZZ for an oil pressure malfunction and then [requested priority handling] once we shut the engine down. The FO remained the pilot flying until the Shutdown Checklist was complete to the deferred items. At that point; I became the pilot flying and flew a flaps 20 single engine approach and landing to [Runway] XXL. We cleared the runway onto Taxiway XX so the fire department could check for any leaking fluids or other hazards. None were found so we taxied under our own power to Gate XX.[The cause was] mechanical failure. Maintenance found evidence of an oil leak at the scavenge pump.