The weed is probably Veronica filiformis--normally a spring or winter annual.
I seeded a nearby thin, burned out area with Pennington "Sun and Shade" which they call a "Premium grass seed mix..." which contains 64 percent perennial ryegrass. Half of it was raked three times and seeded, and the second area was not raked at all--but sowed with three times the usual amount of seed. I counted about 40 seeds per square inch. Neighbor does not sprinkle his lawn. Stay tuned.