论文标题
扩大筛子的总筛和模式
Expanding total sieve and patterns in primes
论文作者
论文摘要
令$ \ big（\ mathcal {s} _n^{α，κ，\ mathfrak {r}}（z）（z）\ big）_ {n = 1}^\ infty $是最大可能的无数次间隔的顺序$ z \ in \ Mathcal {s} _n^{α，κ，\ Mathfrak {r}}}}（z）\ subset \ edrowline {\ mathcal {\ Mathcal {m}} _ n^{α，κ，κ，\ Mathfrak {r}} = \ big big big big big big big big big big big big big big big big big big big big big big _} [\ Mathfrak {r} _i] _ {\ Mathfrak {p} _i} $或$ \ Mathcal {s} _n^{α，α，κ，\ Mathfrak {r}}}}（z）（z）= \ emptyset = \ emptyset $，wher i/κ\ right \ rceil-1} $和$ z \ in \ mathbb {z} $。我们证明了$ \ big（\＃\ Mathcal {s} _n^{α，κ，\ Mathfrak {r}}}（z）（z）\ big）_ {n = 1}^\ infty $ uscillty $ suscillate无限地振荡了很多次，左右左右$α\ in \ mathbb {z}^+$，$κ\ in \ mathbb {z} \ cap [1，p_α）$和$ \ mathfrak {r} _i \ in \ mathbb {z} $。令$ t =（a_1，a_2，\ ldots，a_k）$是可允许的$ k $ -tuple，然后让$ \ MATHCAL {x} _n^{t，k，ρ，η} = \ left \ {x \ in [ρ]_η\，：\，\，\ {x \！+\！a_1，x \！ _2，\ ldots，x \！+\！a_k \} \ cap \ mathcal {m} _ {n+α-1} \ neq \ neq \ emptyset \ right \} $对于\ Mathbb {z}^+$中的每个$ n \，其中$ \ mathcal {m} _g = \ bigCup_ {i = 1}^g [0] _ {p_i} $。我们证明，对于任何$ t $，对于某些固定的$α$，$κ$，$ρ$，$η$，$ z $和$ \ mathfrak {r} $，存在$ \ overline {\ mathcal {\ mathcal {m}}}} _ {κN} _ {κN}^$ nine { $ \ Mathcal {x} _n^{t，k，ρ，η} $ in \ mathbb {z}^+$。这意味着任何扩展的整数间隔的长度都被$ \ Mathcal {m} _ {n+α-1} $ coscillates筛分出了$ t $的所有发生的长度。 Eratosthenes筛子的概念断言$ \ Mathcal {e} _n = [2，p^2_ {n+α}）\ cap \ left（\ Mathbb {z} \ setMinus \ setMinus \ Mathcal \ Mathcal {M Mathcal {M}因此，让$ p^2_ {n+α} =ω\ left（n^2 \ right）$，我们得到$ \ mathcal {e} _n $包括一个符合$ t $ $ n $值的子集，因此，$ n $的$ t $ t $与Prime sequence sequence cone seque coptionce usection $ n $相匹配。
Let $\big(\mathcal{S}_n^{α,κ,\mathfrak{r}}(z)\big)_{n=1}^\infty$ be a sequence of the largest possible integer intervals, such that $z\in\mathcal{S}_n^{α,κ,\mathfrak{r}}(z)\subset\overline{\mathcal{M}}_n^{α,κ,\mathfrak{r}}=\bigcup_{i=1}^n [\mathfrak{r}_i]_{\mathfrak{p}_i}$ or $\mathcal{S}_n^{α,κ,\mathfrak{r}}(z)=\emptyset$, where $\mathfrak{p}_i=p_{α+\left\lceil i/κ\right\rceil-1}$ and $z\in\mathbb{Z}$. We prove that $\big(\#\mathcal{S}_n^{α,κ,\mathfrak{r}}(z)\big)_{n=1}^\infty$ oscillates infinitely many times around $β_n\!=\!o\left(n^2\right)$ for any fixed $α\in\mathbb{Z}^+$, $κ\in\mathbb{Z}\cap[1,p_α)$, and $\mathfrak{r}_i\in\mathbb{Z}$. Let $T=(a_1,a_2,\ldots,a_k)$ be an admissible $k$-tuple and let $\mathcal{X}_n^{T,k,ρ,η}=\left\{x\in[ρ]_η\,:\,\{x\!+\!a_1,x\!+\!a_2,\ldots,x\!+\!a_k\}\cap\mathcal{M}_{n+α-1}\neq\emptyset\right\}$ for each $n\in\mathbb{Z}^+$, where $\mathcal{M}_g=\bigcup_{i=1}^g [0]_{p_i}$. We prove that for any $T$ and for some fixed $α$, $κ$, $ρ$, $η$, $z$, and $\mathfrak{r}$, there exists a linear bijection between $\overline{\mathcal{M}}_{κn}^{α,κ,\mathfrak{r}}$ and $\mathcal{X}_n^{T,k,ρ,η}$ for each $n\in\mathbb{Z}^+$. It implies that the length of any expanding integer interval on which all occurrences of $T$ are sieved out by $\mathcal{M}_{n+α-1}$ oscillates infinitely many times around $\widetildeβ_n=o\left(n^2\right)$. The concept of the sieve of Eratosthenes asserts $\mathcal{E}_n=[2,p^2_{n+α})\cap\left(\mathbb{Z}\setminus\mathcal{M}_{n+α-1}\right)\subset\mathbb{P}$. Therefore, having $p^2_{n+α}=ω\left(n^2\right)$, we obtain that $\mathcal{E}_n$ includes a subset matched to $T$ for infinitely many values of $n$ and, consequently, $T$ matches infinitely many positions in the sequence of primes.
