> But why is it important that these zeros are on the vertical line Im z = 0.5 Great question. The formula works whether or not the zeros have real part 0.5, b

> But why is it important that these zeros are on the vertical line Im z = 0.5 Great question. The formula works whether or not the zeros have real part 0.5, but its implications are different. If you're counting primes up to x you will get the main term (roughly x/log x) and each zero p will give an oscillating error term ~ x^p/p. If the real part is always 0.5 then these error terms are all on sqrt(x) scale, which is what you would expect from random variation. If on the other hand you have a single zero right of the line, say at real part 3/4 (and mirror image one at real part 1/4) then now you get a secondary error term of size x^(3/4) that dominates all of the other error terms, giving some extra structure to the primes: they're no longer random, you can predict where they are more or less likely.