arXiv 1508.073672015-08math.NT

Zeta函数与具有某些特殊素数集的渐近可加基

Zeta functions and asymptotic additive bases with some unusual sets of primes

William D. Banks

备注：: 13 pages; more references added; a few remarks added to Section 1.4

摘要：

修复$δ\in(0,1]$、$σ_0\in[0,1)$和实值函数$\varepsilon(x)$，其中$\limsup_{x\to\infty}\varepsilon(x)\le 0$。对于每一组素数${\mathcal P}$，其计数函数$π_{\mathcal P}(x)$满足$$π_{\mathcal P}(x)=δ\,π(x)+O\bigl(x^{σ_0+\varepsilon(x)}\bigr),$$形式的估计，我们定义一个zeta函数$ζ_{\mathcal P}(s)$，它与黎曼zeta函数$ζ(s)$密切相关。对于$σ_0\le\frac12$，我们证明了黎曼假设等价于$ζ_{\mathcal P}(s)$在区域$\{σ>\frac12\}$中的非零性。对于每一个包含素数$2$且其计数函数满足$$π_{\mathcal P}(x)=δ\,π(x)+O\bigl((\log\log x)^{\varepsilon(x)}\bigr),$$形式的估计的素数${\mathcal P}$的集合，我们证明了${\mathcal P}$是${\mathbb N}$的渐近可加基，即，对于某个整数$h=h({\mathcal P})>0$，和集$h{\mathcal P}$包含除了100个自然数之外的所有自然数。例如，${\mathbb N}$的渐近加法基由集合$$ \{2,547,1229,1993,2749,3581,4421,5281\ldots\}, $$提供，该集合由$2$和其后的每百分之一个素数组成。

英文摘要：

Fix $δ\in(0,1]$, $σ_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $π_{\mathcal P}(x)$ satisfies an estimate of the form $$π_{\mathcal P}(x)=δ\,π(x)+O\bigl(x^{σ_0+\varepsilon(x)}\bigr),$$ we define a zeta function $ζ_{\mathcal P}(s)$ that is closely related to the Riemann zeta function $ζ(s)$. For $σ_0\le\frac12$, we show that the Riemann hypothesis is equivalent to the non-vanishing of $ζ_{\mathcal P}(s)$ in the region $\{σ>\frac12\}$. For every set of primes ${\mathcal P}$ that contains the prime $2$ and whose counting function satisfies an estimate of the form $$π_{\mathcal P}(x)=δ\,π(x)+O\bigl((\log\log x)^{\varepsilon(x)}\bigr),$$ we show that ${\mathcal P}$ is an asymptotic additive basis for ${\mathbb N}$, i.e., for some integer $h=h({\mathcal P})>0$ the sumset $h{\mathcal P}$ contains all but finitely many natural numbers. For example, an asymptotic additive basis for ${\mathbb N}$ is provided by the set $$ \{2,547,1229,1993,2749,3581,4421,5281\ldots\}, $$ which consists of $2$ and every hundredth prime thereafter.

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摘要：

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