The Goldbach exceptional set: an exponent 7/12 and conditional refinement
Abstract
Let E(X) count the even integers between 4 and X that are not sums of two primes. We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwiłł to prove E(X)≪ε X7/12+ε whenever all Dirichlet L-functions have a common fixed zero-free half-plane Rea s>θ0, with θ0<1. The uniform 7/8 theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most θ, where 1/2≤θ≤2/3, we prove E(X)≪θ,εXg(θ)+ε, g(θ)=(θ(5−4θ))/(2(2−θ)). Thus every fixed θ<2/3 gives an exponent strictly below 7/12. The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.
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BibTeX
@misc{pintz2026goldbach,
title = {{The Goldbach exceptional set: an exponent 7/12 and conditional refinement}},
author = {János Pintz and Ingo Althöfer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15559},
eprint = {2610.15559},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15559},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Goldbach exceptional set: an exponent 7/12 and conditional refinement
AU - János Pintz
AU - Ingo Althöfer
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15559
UR - https://arxiv.org/abs/2610.15559
AB - Let E(X) count the even integers between 4 and X that are not sums of two primes. We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwiłł to prove E(X)≪_ε X^7/12+ε whenever all Dirichlet L-functions have a common fixed zero-free half-plane Rea s>θ_0, with θ_0<1. The uniform 7/8 theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most θ, where 1/2≤θ≤2/3, we prove E(X)≪_θ,εX^g(θ)+ε, g(θ)=(θ(5−4θ))/(2(2−θ)). Thus every fixed θ<2/3 gives an exponent strictly below 7/12. The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.
ER -