The Riemann zeta function ζ(s) has no zeros in the half-plane Re(s) > 1, and the Riemann Hypothesis asserts that all nontrivial zeros lie on Re(s) = 1/2. Between these extremes, zero-free regions — subsets of the critical strip proved to contain no zeros — are the working currency of analytic number theory.
The paper on zero-free regions inspired by Heath-Brown (arXiv: 2603.21490) establishes a new explicit zero-free region: ζ(s) ≠ 0 when t ≥ 3 and σ ≥ 1 − 1/(4.896 log t).
The constant 4.896 improves on previous explicit bounds. The method draws on Heath-Brown's work on Linnik's constant — the constant L such that the least prime in an arithmetic progression a mod q is at most q^L. Zero-free regions and Linnik's constant are entangled: better zero-free regions produce smaller Linnik constants, and techniques developed for one problem inform the other.
The explicit nature matters. Classical zero-free regions are “sufficiently large t” theorems. Explicit versions — with computable constants valid for specific ranges — are what's needed for computational number theory: verifying the Riemann Hypothesis up to a height, computing explicit bounds on prime-counting functions, certifying primality tests.
The through-claim: an explicit constant is a different kind of result from an asymptotic one. The zero-free region σ ≥ 1 − 1/(c log t) is classical for some unspecified c. Making c = 4.896 explicit, valid for t ≥ 3, transforms a qualitative statement into a computable bound. The mathematics is the same; the utility is fundamentally different.
2603.21490. Number theory / Riemann zeta function / zero-free region / Linnik's constant / explicit bounds.