LinnikSetups

LinnikSetups are a conceptual framework in analytic number theory used to organize and standardize proof strategies for bounding the least prime in an arithmetic progression. The term acknowledges Linnik's theorem, which asserts the existence of a bound p ≤ q^L for primes congruent to a modulo q, with the exponent L known as the Linnik constant. A LinnikSetup provides a structured sequence of analytic steps, aiming to optimize the dependence on the modulus q and to make the use of auxiliary tools transparent.

The core components typically include Dirichlet characters modulo q, L-functions and their zero-free regions, zero-density estimates, and bilinear form or exponential sum methods. A LinnikSetup also prescribes the use of smoothing techniques, parameter choices, and the treatment of potential exceptional zeros, all arranged to yield an effective bound for the least prime. The framework emphasizes modularity, so different estimates can be swapped in as sharper results become available, while preserving the overall structure of the argument.

An outline of a typical LinnikSetup might involve: selecting a suitable modulus q and residue a with (a, q) = 1; expressing primes via Dirichlet characters and their L-functions; applying zero-free region results and zero-density bounds to control character sums; employing smoothing and bilinear form estimates to optimize the dependence on q; and extracting a concrete bound p ≤ q^L with an explicit or implicit L depending on the chosen tools. The goal is to provide a reusable, transparent blueprint that can be adapted to variations of the problem.

See also Linnik's theorem, Dirichlet L-functions, zero-density estimates, and primes in arithmetic progressions.