Siegel零点存在时的Hardy-Littlewood-Chowla猜想
The Hardy--Littlewood--Chowla conjecture in the presence of a Siegel zero
摘要：
假设Siegel零存在，我们证明了一个混合版本的Chowla和Hardy-Littlewood素数元组。因此，对于自然数的无穷序列$x$和任何不同的整数$h_1,\dots,h_k,h'_1,\dots,h'_\ell$，我们建立了$$\sum_{n\leq x}Λ(n+h_1)\cdots Λ(n+h_k)λ(n+h_{1}')\cdots λ(n+h_{\ell}')$$对任何$0\leq k\leq 2$和$\ell \geq 0$的渐近公式。专门针对$\ell=0$或$k=0$，我们推出了以前已知的结果Hardy-Littlewood（或孪生素数）猜想和Chowla猜想的Siegel零点的存在下，由于Heath-Brown和Chinis，分别。我们的渐近公式的有效性范围比这些以前的结果更广泛。
英文摘要：
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 $$\sum_{n\leq x}Λ(n+h_1)\cdots Λ(n+h_k)λ(n+h_{1}')\cdots λ(n+h_{\ell}')$$ 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.
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