On the Second Hardy-Littlewood Conjecture
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913718527328256 |
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| author | Chahal, Bittu Elma, Ertan Fellini, Nic Vatwani, Akshaa Vo, Do Nhat Tan |
| author_facet | Chahal, Bittu Elma, Ertan Fellini, Nic Vatwani, Akshaa Vo, Do Nhat Tan |
| contents | The second Hardy-Littlewood conjecture asserts that the prime counting function $π(x)$ satisfies the subadditive inequality
\begin{align*}
π(x+y)\leqslant π(x)+π(y)
\end{align*} for all integers $x,y\geqslant 2$. By linking the subadditivity of $π(x)$ to the error term in the Prime Number Theorem, we obtain unconditional improvements on the range of $y$ for which $π(x)$ is known to be subadditive. Moreover, assuming the Riemann Hypothesis, we show that for all $ε>0$, there exists $x_ε \geqslant 2$ such that for all $x\geqslant x_ε$ and $y$ in the range \begin{align*}
\frac{(2+ε)\sqrt{x}\log^2x}{8π}\leqslant y\leqslant x, \end{align*} the inequality $π(x+y)\leqslant π(x) + π(y)$ holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02766 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Second Hardy-Littlewood Conjecture Chahal, Bittu Elma, Ertan Fellini, Nic Vatwani, Akshaa Vo, Do Nhat Tan Number Theory 11N05 The second Hardy-Littlewood conjecture asserts that the prime counting function $π(x)$ satisfies the subadditive inequality \begin{align*} π(x+y)\leqslant π(x)+π(y) \end{align*} for all integers $x,y\geqslant 2$. By linking the subadditivity of $π(x)$ to the error term in the Prime Number Theorem, we obtain unconditional improvements on the range of $y$ for which $π(x)$ is known to be subadditive. Moreover, assuming the Riemann Hypothesis, we show that for all $ε>0$, there exists $x_ε \geqslant 2$ such that for all $x\geqslant x_ε$ and $y$ in the range \begin{align*} \frac{(2+ε)\sqrt{x}\log^2x}{8π}\leqslant y\leqslant x, \end{align*} the inequality $π(x+y)\leqslant π(x) + π(y)$ holds. |
| title | On the Second Hardy-Littlewood Conjecture |
| topic | Number Theory 11N05 |
| url | https://arxiv.org/abs/2503.02766 |