I agree. This integral cannot be constantly zero in that region, as the integrand is not itself constantly zero. Also, there's no reason to expect this integra

I agree. This integral cannot be constantly zero in that region, as the integrand is not itself constantly zero. Also, there's no reason to expect this integral (merely the last term of the explicit formula, the one arising from the trivial zeros of the zeta function) to count primes all in its own, so no reason it should be zero on its own below 2. That statement instead correctly describes the sum of all four terms, in the article's presentation.