arXiv 1903.054872019-03math.NT

素分圆域Euler-Kronecker常数的有效计算

Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields

Alessandro Languasco

备注：: 25 pages, 6 tables, 4 figures. Third known example of negative values for Ek(q) inserted. Complete set of computation of Ek(q) and Ek(q)^+ for every prime up to 10^6; computation of max|L'/L(1,chi)| for the same primes inserted. Two references added, typos corrected
发表信息：: Research in Number Theory 7 (2021), 1--22
DOI：: 10.1007/s40993-020-00213-1

摘要：

本文提出了一种新的计算素分圆域$\mathbb{Q}(ζ_q)$的Euler-Kronecker常数$\mathfrak{G}_q$的算法，其中$q$是奇素数，$ζ_q$是本原$q$ -单位根.利用这种新算法，我们计算了$\mathfrak{G}_q$和$\mathfrak{G}_q^+$，其中$\mathfrak{G}_q^+$是$\mathbb{Q}(ζ_q)$的最大实子域的Euler-Kronecker常数，对于一些非常大的素数$q$，从而得到了两个新的$\mathfrak{G}_q$的负值：$\mathfrak{G}_{9109334831}= -0.248739\dotsc$和$\mathfrak{G}_{9854964401}= -0.096465\dotsc$。我们还计算了$\mathfrak{G}_q$和$\mathfrak{G}^+_q$，对于每个奇素数$q\le 10^6$，从而扩大了以前已知的$\mathfrak{G}_q$和$\mathfrak{G}^+_q$的范围。我们的方法还表明，差值$\mathfrak{G}_q - \mathfrak{G}^+_q$的计算方法比它的两个被加数都要简单得多，见3.4节。此外，作为一个副产品，我们还计算了每一个奇素数$q\le 10^6$的$M_q=\max_{χ\ne χ_0} \vert L^\prime/L(1,χ) \vert $，其中$L(s,χ)$是Dirichlet $L$ -函数，$χ$运行在非平凡Dirichlet特征标mod $q$上，$χ_0$是平凡Dirichlet特征标mod $q$上.作为我们计算的另一个副产品，我们还将提供更多关于算术级数中广义欧拉常数的数据。用于执行此处描述的计算的程序和获得的数值结果可在以下网址获得：\url{http://www.math.unipd.it/~languasc/EK-comput.html}。

英文摘要：

We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler-Kronecker constants $\mathfrak{G}_q$ for the prime cyclotomic fields $\mathbb{Q}(ζ_q)$, where $q$ is an odd prime and $ζ_q$ is a primitive $q$-root of unity. With such a new algorithm we evaluated $\mathfrak{G}_q$ and $\mathfrak{G}_q^+$, where $\mathfrak{G}_q^+$ is the Euler-Kronecker constant of the maximal real subfield of $\mathbb{Q}(ζ_q)$, for some very large primes $q$ thus obtaining two new negative values of $\mathfrak{G}_q$: $\mathfrak{G}_{9109334831}= -0.248739\dotsc$ and $\mathfrak{G}_{9854964401}= -0.096465\dotsc$ We also evaluated $\mathfrak{G}_q$ and $\mathfrak{G}^+_q$ for every odd prime $q\le 10^6$, thus enlarging the size of the previously known range for $\mathfrak{G}_q$ and $\mathfrak{G}^+_q$. Our method also reveals that difference $\mathfrak{G}_q - \mathfrak{G}^+_q$ can be computed in a much simpler way than both its summands, see Section 3.4. Moreover, as a by-product, we also computed $M_q=\max_{χ\ne χ_0} \vert L^\prime/L(1,χ) \vert $ for every odd prime $q\le 10^6$, where $L(s,χ)$ are the Dirichlet $L$-functions, $χ$ run over the non trivial Dirichlet characters mod $q$ and $χ_0$ is the trivial Dirichlet character mod $q$. As another by-product of our computations, we will also provide more data on the generalised Euler constants in arithmetic progressions. The programs used to performed the computations here described and the numerical results obtained are available at the following web address: \url{http://www.math.unipd.it/~languasc/EK-comput.html}.

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。

摘要：

英文摘要：

相关论文