After Lyra and Will obtain the subtle knife, Tullio, the young man who stole the knife in order to ward off Specters, is attacked and exhibits some odd behavior:
Then he [Tullio] turned away and began to run his hands along the stones in the wall, looking closely at each one, counting them, feeling the edges, hunching up his shoulders as if to ward off something behind him, shaking his head.
Lyra witnesses this, and describes it to Will later on:
"He started counting the stones in the wall. He sort of felt all over them... But he couldn't keep it up. In the end he sort of lost interest and stopped. Then he was just still ... Why?"
"Because ... I think maybe they come from my world after all, the Specters. If they make people behave like that, I wouldn't be surprised at all if they came from my world. ... Maybe they're not called Specters [in Will's world]. Maybe we call them something else."
This is not the (full) origin of the Specters, but this isn't revealed until much later. Will was comparing this behavior to some of the odd behavior of his mother, who exhibited some similar behavior in an attempt to drive off some real or imagined threat.
As far as I can tell, this behavior of Specter victims is not mentioned again, nor why Will thought his mother had specifically been attacked by a Specter (or something similar), nor how this counting behavior was supposed to protect against Specters in any way.
What is the significance of these passages? It feels odd for it to be highlighted twice, and then just forgotten. Was Phillip Pullman (through Will) referring to some specific real-world condition, or comparing Specters to a mythological creature?