论文标题
快速算法来计算Ramanujan-Deninger伽马功能和一些数字理论应用
A fast algorithm to compute the Ramanujan-Deninger gamma-function and some number-theoretic applications
论文作者
论文摘要
我们引入了一种快速算法，以计算Ramanujan-deningerγ函数及其在正值下的对数衍生物。这样的算法使我们能够极大地扩展有关Euler-Kronecker的数值调查，$ \ Mathfrak {G} _Q $，$ \ Mathfrak {G} _Q^+$和$ M_Q = \ \ \ \ \ axmax_max_max_ {χ\ neχ_0χ_0} \ vert l^$ prime Is $ $ prime $ $ Prime，$χ$在原始dirichlet字符上运行$ \ bmod \ q $，$χ_0$是琐碎的dirichlet carture $ \ bmod \ q $和$ l（s，χ）$是dirichlet $ l $ l $ function $ l $ unction $χ$。使用这样的算法，我们获得了$ \ mathfrak {g} _ {50 040 955 631} = -0.16595399 \ dotsc $和$ \ Mathfrak {g} _ {g} _ {50 040 955 631} $ \ mathfrak {g} _q $。此外，我们还计算$ \ mathfrak {g} _q $，$ \ mathfrak {g} _q^+$和$ m_q $对于每个奇数$ q $，$ 10^6 <q \ le 10^7 $，从而取得了先前的结果。结果，我们得到的是，$ \ mathfrak {g} _q $和$ \ mathfrak {g} _q^+$对于每一个奇数prime $ q $最高$ 10^7 $都是阳性Q \ le 10^7 $。实际上，下限为$ q> 13 $。所述的程序和此处描述的结果是在以下地址\ url {http://www.math.unipd.it/~languasc/somcomp-appl.html}中收集的。
We introduce a fast algorithm to compute the Ramanujan-Deninger gamma function and its logarithmic derivative at positive values. Such an algorithm allows us to greatly extend the numerical investigations about the Euler-Kronecker constants $\mathfrak{G}_q$, $\mathfrak{G}_q^+$ and $M_q=\max_{χ\ne χ_0} \vert L^\prime/L(1,χ)\vert$, where $q$ is an odd prime, $χ$ runs over the primitive Dirichlet characters $\bmod\ q$, $χ_0$ is the trivial Dirichlet character $\bmod\ q$ and $L(s,χ)$ is the Dirichlet $L$-function associated to $χ$. Using such algorithms we obtained that $\mathfrak{G}_{50 040 955 631} =-0.16595399\dotsc$ and $\mathfrak{G}_{50 040 955 631}^+ =13.89764738\dotsc$ thus getting a new negative value for $\mathfrak{G}_q$. Moreover we also computed $\mathfrak{G}_q$, $\mathfrak{G}_q^+$ and $M_q$ for every odd prime $q$, $10^6< q\le 10^7$, thus extending previous results. As a consequence we obtain that both $\mathfrak{G}_q$ and $\mathfrak{G}_q^+$ are positive for every odd prime $q$ up to $10^7$ and that $\frac{17}{20} \log \log q< M_q < \frac{5}{4} \log \log q $ for every odd prime $1531 < q\le 10^7$. In fact the lower bound holds true for $q>13$. The programs used and the results here described are collected at the following address \url{http://www.math.unipd.it/~languasc/Scomp-appl.html}.
