According to Atticus, what is the state prosecutor counting on to convince the jury to convict Tom Robinson in To Kill a Mockingbird?
I think that you are talking about what Atticus says in Chapter 20 when he is delivering his closing argument to the jury. He does not really focus so much on the prosecutor as on the Ewells, though.
What he says they are relying on is a prejudice against blacks. He says they are relying on the jury to believe that blacks are completely immoral and not to be trusted. He says that their testimony requires the jury to believe that all black people lie all the time and that all black men want to violate white women if only they could get the chance.
Atticus is quite aware from the start of To Kill a Mockingbird that no all-white jury is likely to take the word of a black man over that of a white man. The prosecutor, Horace Gilmer, also recognizes this fact, and he never fails to remind the jury of Tom's color, calling him "boy" repeatedly during his cross-examination. Atticus also mentions the assumption that many white people believe that "all Negroes lie, that all Negroes are basically immoral human beings."