数域上的默滕斯第三定理：一种新证明、克拉梅尔不等式、振荡与偏差
Mertens' Third Theorem for Number Fields: A New Proof, Cramér's Inequality, Oscillations, and Bias
AI总结：
本文给出了数域上默滕斯第三定理的新证明，证明了误差项在技术条件下无穷多次变号所需的数域克拉梅尔不等式，推广了拉姆祖里关于误差项为正的对数密度结果，并对两个二次域进行了数值研究。
摘要：
本文的第一个结果是数域情形下默滕斯第三定理的另一种证明，该证明推广了哈代的一种方法。第二个结果涉及默滕斯第三定理中误差项的符号。戴蒙德和平茨证明了经典情形下误差项无穷多次变号，本文在假设一个合理的技术条件下为数域建立了这一结果。为此，我们需要证明数域上的克拉梅尔不等式，这本身也具有重要意义。拉姆祖里在戴蒙德和平茨工作的基础上证明了使得默滕斯第三定理误差项为正的实数 $x \ge 2$ 的集合具有对数密度，因此本文的第三个结果将拉姆祖里的结果推广到数域情形。我们还基于鲁宾斯坦和萨尔纳克在经典情形下所做的类似工作，对数域 $\mathbb{Q}(\sqrt{5})$ 和 $\mathbb{Q}(\sqrt{13})$ 进行了数值研究。
英文摘要：
The first result of our article is another proof of Mertens' third theorem in the number field setting, which generalises a method of Hardy. The second result concerns the sign of the error term in Mertens' third theorem. Diamond and Pintz showed that the error term in the classical case changes sign infinitely often and in our article, we establish this result for number fields assuming a reasonable technical condition. In order to do so, we needed to prove Cramér's inequality for number fields, which is interesting in its own right. Lamzouri built upon Diamond and Pintz's work to prove the existence of the logarithmic density of the set of real numbers $x \ge 2$ such that the error term in Mertens' third theorem is positive, so the third result of our article generalises Lamzouri's results for number fields. We also include numerical investigations for the number fields $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{13})$, building upon similar work done by Rubinstein and Sarnak in the classical case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
