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arXiv · 2605.19566

On the Goldbach problem with restricted primes

Michael Harm
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Abstract

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density estimate for Dirichlet $L$-functions, we may unconditionally take $U= N^{\frac{4}{49}}\exp(\log^{\frac{2}{3}+\varepsilon}N)$ for any $\varepsilon>0$. If we assume the Generalized Riemann Hypothesis instead, we may take $U= \log^{4+\varepsilon}N$.

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Michael Harm. 2026-05-19. On the Goldbach problem with restricted primes. https://arxiv.org/abs/2605.19566

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