B757 Captain reported a pressurization failure and divert to a nearby airport.

Narrative

During step climb from 34000 ft. to 38000 ft. the L Autopilot disconnected. We silenced the warning flew the plane manually while reconnecting the Autopilot without incident. Approximately 10 to 15 minutes later the Warning Siren sounded again and after initially suspecting another Autopilot disconnect; we observed Cabin Altitude Warning lights and EICAS message. The Cabin Altitude indicated above 10000 ft. and the Outflow Valve indicated fully closed. We dawned our O2 masks; established communication; advised ATC we needed a lower altitude and initiated the Cabin Altitude checklist from the QRC. ATC issued a Heading and cleared our descent to 10000 ft. Established on the assigned heading and descending in FLCH; we continued the Checklist by voice and hand signals. I refined the descent rate by increasing the airspeed and extending the speed brakes; while no definitive control of the Cabin Altitude seemed to be available through the checklist items.I then transferred the controls to the First Officer so I could continue the checklist with the iPad; plan for a diversion and attempt to contact Dispatch. We coordinated with ATC to [request priority handling] and make a turn towards ZZZ; as this appeared to be the most suitable airport available at the time. I finished the Cabin Altitude Checklist with the iPad and then consulted the Cabin Auto Inop Non-Normal for reference. At this point I noted that the Cabin Altitude Indicator pointed to OFF. While manual control of the Out Flow Valve appeared to be functional; the OFF indication of the Cabin Altitude was somewhat confusing and I elected to manually set the outflow valve to a 500 FPM Climb and work on planning and coordination for the diversion with Dispatch. Dispatch was contacted and concurred with the plan to divert to ZZZ. Once the Cabin Altitude was safely below 10000 ft. I removed my oxygen mask and contacted the flight attendants to check on the status of the cabin and provide them with the briefing items. They reported no abnormalities in the cabin; the passenger supplemental oxygen masks had not deployed and all passengers were down and secure. I then briefed the passengers over the PA that we had experienced a malfunction with our Pressurization System and were making a precautionary landing in ZZZ. During descent below 10000 ft. the Cabin Altitude indicator was still reading OFF; but other indications appeared relatively normal and we made a plane to check that the outflow valve was full open after landing and the cabin was unpressurized. The First Officer (FO) and I then setup and briefed the approach; I resumed Pilot Flying duties; we completed a Descent Checklist and made an uneventful landing in ZZZ.