How do you estimate global species richness?
You might be interested in the below works as well as the references that they cite.

May, R. M. (1988). How many species are there on Earth? Science, 241(4872), 1441–1449.

Millennium Ecosystem Assessment. (2005). Ecosystems and Human Well-Being: Synthesis (p. 137). Washington, D.C.: Island Press.

It's worth noting that many estimates of species richness at large scales are based on the use of the species-area relationship to upscale richness measurements from small scales to large scales.