Outlets in living room suddenly died but there is not tripped circuit breaker
Non-contact voltage testers can be very sensitive and misleading. So don't worry too much about that. 

This may have been asked before but do ANY of the outlets on the breaker that controls the dead ones have power? If you do, there is probably a connection failure in one of the outlets downstream.   That doesn't eliminate the first outlet in the circuit having the failure though, so it's not a perfect test. 

2 things come to mind:  1) The infamous "back stab" outlets that frequently fail. You should get a real multi meter and test between hot and neutral, hot and ground, and neutral and ground.  If  you have power from neutral to ground, you have an open neutral someplace.  Multi-meters can be very sensitive as well as non contact testers. It's often useful to have some sort of a load plugged in (such as your lamp), to eliminate any spurious, induced voltages. 

With the POWER OFF (Check with meter or non-contact tester to be sure), you might want to start removing the outlets and inspect for damage, loose connections, etc.   

2)  I had this issue in a previous house I lived in and it turned out to be rodent damage.  A mouse had chewed thru a cable.  First breaking the circuit and then electrocuting himself!  I know because when I opened up the sheetrock there was his little fried body.