论文标题
在beurling质量定理中剩余项的振荡，“由给定的Zeta-Zero引起”
Oscillation of the remainder term in the prime number theorem of Beurling, "caused by a given zeta-zero"
论文作者
论文摘要
继续对Beurling Zeta函数进行的先前研究，在这里我们证明了两个结果，从而推广了有关Riemann Zeta功能的经典案例的长期知识及其概括。 首先，我们解决了利特伍德（Littlewood）的问题，他要求明确的振荡结果，只要知道Zeta-Zero。我们证明，给定的$ρ_0$的beurling zeta函数$ζ_p$对于由Primes $ p $生成的给定数字系统，相应的错误项$δ_p（x）：= =ψ_{p}（x）（x）-x $ $（π/2- \ varepsilon）x^{\ reρ_0}/|ρ_0| $。 该研究中解释了常数$π/2 $的神秘外观。最后，我们证明是本文的下一个主要结果：给定的$ \ varepsilon> 0 $，存在带有Primes $ p $的beurling编号系统，因此$ |δ_p（x）| \ le（π/2+\ varepsilon）x^{\ reρ_0}/|ρ_0| $。 在第二部分中，通过应用Broucke和Vindas的奇妙素数随机近似结果，使低规范的正弦多项式的非平凡结构耦合在一起，Broucke和Vindas的近期随机近似结果促进了由于钻石，蒙哥马利和Vorhauer而引起的突破性概率结构。
Continuing previous study of the Beurling zeta function, here we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $ρ_0$ of the Beurling zeta function $ζ_P$ for a given number system generated by the primes $P$, the corresponding error term $Δ_P(x):=ψ_{P}(x)-x$, where $ψ_{P}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as $(π/2-\varepsilon) x^{\Re ρ_0}/|ρ_0|$. The somewhat mysterious appearance of the constant $π/2$ is explained in the study. Finally, we prove as the next main result of the paper the following: given $\varepsilon>0$, there exists a Beurling number system with primes $P$, such that $|Δ_P(x)| \le (π/2+\varepsilon)x^{\Re ρ_0}/|ρ_0|$. In this second part a nontrivial construction of a low norm sine polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery and Vorhauer.
