Anomaly-Driven Sieve: Uncovering Exceptional Sets in Prime Distributions

Traditional sieve methods, as surveyed in Greaves, Harman, and Huxley (1997) and Tenenbaum (1995), are designed to filter integers according to expected regularities in prime occurrence. However, little systematic attention has been paid to cataloguing and understanding the sets where these expectations break down—those "exceptional" or anomalous sets. This project proposes to build a new class of sieves that, rather than discarding or minimizing anomalies, explicitly targets and classifies them. By quantifying and mapping the nature and density of these exceptional sets, we could potentially uncover new structures or conjectures about prime distribution, offering a sharp tool for investigating unsolved questions like the existence and distribution of large prime gaps or clusters. This idea builds directly on the “Investigate deviations from expectations” heuristic and could lead to a new taxonomy of exceptions in additive number theory.

References:

1. Sieve methods, exponential sums, and their applications in number theory : proceedings of a symposium held in Cardiff, July 1995. G. Greaves, G. Harman, M. Huxley (1997).
2. Introduction to Analytic and Probabilistic Number Theory. G. Tenenbaum (1995).