How the Zeros of the Zeta Function Predict the Distribution of Primes

Snapshot 1: the graphs of and with no correction term

Snapshot 2: the graphs of and with correction term that uses the first 20 pairs of zeros of the zeta function

Snapshot 3: a closeup showing that with correction term correctly predicts the primes 103, 107, 109, 113, and 127, and the gap between 113 and 127; notice that, with no correction term, by itself misses these details

(Mathematica's PrimePi[x]) is a step function that jumps up by 1 whenever is prime. Riemann's prime-counting function (RiemannR[x], introduced in Mathematica version 7) is defined as .

is a good approximation to π(x). Unfortunately, does not follow the jumps in . Furthermore, when a gap occurs in the primes, (for example, between 23 and 29), keeps increasing even though remains constant.

However, if we add to a correction term that uses the first few zeros of the Riemann zeta function, something very surprising happens: we get a new function that very closely matches the jumps and irregularities of ! When is prime, this new function increases by about 1 near , so, in effect, it "knows" where the primes are. And when a gap occurs, this new function tends to level out, again emulating the behavior of .

It's almost as if the zeros of the zeta function determine which numbers are prime!

The more zeros we use, the closer we can approximate . For larger , the correction term needs to include more zeros in order to accurately approximate .

The first three zeros are approximately , , and . (The zeros occur in conjugate pairs, so if is a zero, then so is .) The famous Riemann hypothesis is the claim that these "nontrivial" zeros all have real part 1/2.

The correction term for a given is computed as follows. First, let be the smallest integer such that . Define as a sum of terms involving the first pairs of conjugate zeros of the zeta function:

,

where is the exponential integral taking the principal value, and is the nontrivial zeta zero. The imaginary parts of the corresponding terms in cancel. So, becomes

.

Finally, the correction term, which we add to , is .

References:

[1] H. Riesel, Prime Numbers and Computer Methods for Factorization, 2nd ed., Boston: Birkhauser, 1994 pp. 44–55.

[2] H. Riesel and G. Gohl, "Calculations Related to Riemann's Prime Number Formula," Mathematics of Computation, 24(112), 1970 pp. 969–983.

[3] S. Wagon, Mathematica in Action, 2nd ed., New York: Springer, 1999 pp. 540–554.