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The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero
- Source :
- Journal of the London Mathematical Society. 106:3317-3378
- Publication Year :
- 2022
- Publisher :
- Wiley, 2022.
-
Abstract
- Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$, we establish an asymptotic formula for $$\sum_{n\leq x}\Lambda(n+h_1)\cdots \Lambda(n+h_k)\lambda(n+h_{1}')\cdots \lambda(n+h_{\ell}')$$ for any $0\leq k\leq 2$ and $\ell \geq 0$. Specializing to either $\ell=0$ or $k=0$, we deduce the previously known results on the Hardy--Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to Heath-Brown and Chinis, respectively. The range of validity of our asymptotic formula is wider than in these previous results.<br />Comment: 54 pages, no figures. To appear in J. London Math. Soc
Details
- ISSN :
- 14697750 and 00246107
- Volume :
- 106
- Database :
- OpenAIRE
- Journal :
- Journal of the London Mathematical Society
- Accession number :
- edsair.doi.dedup.....cab1aee05163e7ba9843fdcfa9b6f4c5