Unconditional error-term bounds for Chebotarev-based inclusion–exclusion in Artin-type density computations

Develop unconditional effective error-term bounds for prime-counting asymptotics arising from the Chebotarev density theorem, strong enough to carry out the inclusion–exclusion method to compute densities of primes p ≤ x satisfying conditions such as p ≡ 1 (mod n) and a being an n-th power modulo p, without assuming the generalized Riemann hypothesis.

Background

In the discussion of using Chebotarev density to count primes with specified residue and power conditions, the authors point out that effective bounds on error terms are needed to run inclusion–exclusion and obtain densities. They note that such bounds are currently achieved under GRH, and they do not know how to obtain them unconditionally.

To avoid this reliance within the benchmark problem, the authors replace ord_p(a) with a truncated quantity ord_{p, x}(a), which eliminates the need for error-term control, highlighting the unresolved status of unconditional effective error bounds necessary for the full argument.

References

Bounding the error terms is what we need the generalization of Riemann's hypothesis for (c.f.: the standard RH bounds the error term in the prime number theorem); we don't know how to do it without.

— FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI (2411.04872 - Glazer et al., 7 Nov 2024) in Appendix, Section 'Sample problem 1 — high difficulty', Background