Explicit estimates for the Goldbach summatory function
Gautami Bhowmik, Anne-Maria Ernvall-Hytönen, Neea Palojärvi
TL;DR
This work delivers fully explicit, unconditional bounds for the Goldbach summatory function S(x) and its arithmetic-progression analogue S(x;q,a,b) by developing an explicit framework connecting G(n) to Dirichlet L-functions and their zeros. The authors construct two tiers of results: completely explicit numerical bounds and general explicit estimates that sharpen as explicit zero-data improves. Central to the approach are explicit decompositions into sums over zeros, precise control of oscillatory terms with H(x,χ) and R(x;χ1,χ2), and new explicit estimates for ψ(u,χ) that feed into the zero-sum bounds. The results provide concrete error terms, depend only on verifiable zero-free regions and zero-counts, and thus support asymptotic formulas in the GRH regime while remaining unconditional, with direct applicability to primes in arithmetic progressions and to the broader analytic number theory of additive prime problems.
Abstract
In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper, we obtain explicit numerical estimates for the average order of its summatory function both in the classical case and in arithmetic progressions. In addition, we derive new explicit estimates for sums over zeros and for the function $ψ(u,χ)$. Our results are general and describe how the explicit bounds depend on other known explicit estimates. These support the known asymptotic results under the (Generalised) Riemann Hypothesis involving error terms.