Authors establish the Erdős-Graham conjecture for large k and provide GRH-conditional counterexamples for small k using sieves and exponential sums.
Refinements for primes in short arithmetic progressions
2 Pith papers cite this work. Polarity classification is still indexing.
2
Pith papers citing it
abstract
Given a zero-free region and an averaged zero-density estimate over all Dirichlet $L$-functions modulo $q\in\mathbb{N}$, we refine the error terms of the prime number theorem in all and almost all short arithmetic progressions. For example, if we assume the Generalized Density Hypothesis, then for any arithmetic progression modulo $q\leq \log^{\ell} x$ with $\ell>0$ and any $\varepsilon>0$, the prime number theorem holds in all intervals $(x-\sqrt{x}\exp(\log^{\frac{2}{3}+\varepsilon} x),x]$ and almost all intervals $(x-\exp(\log^{\frac{2}{3}+\varepsilon} x),x]$ as $x\rightarrow\infty$. This refines the classic intervals $(x-x^{1/2+\varepsilon},x]$ and $(x-x^\varepsilon,x]$ for any $\varepsilon>0$.
fields
math.NT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Proves asymptotic for the number of ternary Goldbach representations of large odd N with one prime bounded by U = N^{4/49} exp(log^{2/3+ε}N) unconditionally or log^{4+ε}N under GRH.
citing papers explorer
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Binomial coefficients with divisors avoiding an interval
Authors establish the Erdős-Graham conjecture for large k and provide GRH-conditional counterexamples for small k using sieves and exponential sums.
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On the Goldbach problem with restricted primes
Proves asymptotic for the number of ternary Goldbach representations of large odd N with one prime bounded by U = N^{4/49} exp(log^{2/3+ε}N) unconditionally or log^{4+ε}N under GRH.