Near a rational , is governed by how primes distribute in arithmetic progressions mod , which classically ties to the zeros of Dirichlet -functions : the fewer zeros near the line , the more evenly primes are spread and the sharper the major-arc estimate. Rather than rely on classical zero-free regions (which give correct qualitative behavior but hide large, impractical constants), Helfgott used rigorous, computer-verified tables of -function zeros (work of David Platt) to get fully explicit, unconditional bounds — with no need to assume the Generalized Riemann Hypothesis, even though GRH would make the argument shorter.
A second technical innovation on the major arcs is Helfgott's use of Gaussian smoothing weights in place of the sharp cutoffs used in classical treatments; the Mellin transform of a Gaussian-weighted exponential sum brings in parabolic cylinder functions, for which Helfgott had to derive new fully explicit bounds via the saddle-point method, since none were available in the literature. Combining the explicit -function zero data with these smoothed estimates yields a major-arc contribution matching the expected singular series to within an error small enough to be dominated once the minor arcs are controlled — provided .