脆性不变基础

不完全剩余域情形下B_dR^+的伽罗瓦论

Galois theory of B_dR^+ in the imperfect residue field case

本文将Iovita-Zaharescu关于B_dR^+的伽罗瓦论的工作推广到不完全剩余域的情形。证明是基于Colmez的高Kahler微分的结构定理。

Repeated concatenations in residue classes

We give an algorithm to determine all the repeated concatenations, in a given base, of a natural number in a residue class. The author recently describes a particular sequence of $v$-palindromes that inspires this investigation. We also generalize this sequence and discuss how there could be variations of the investigation we do in this paper.

Class Number of the Imaginary Quadratic Field and Quadratic Residues Identities

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for the sum of the quadratic residues are obtained. We finalize the paper by obtaining several identities involving $h(-p)$ the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$

A note on the Voronoi congruences and the residue of the Fermat quotient

Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruence…

基- $q$与Ostrowski数字和在剩余类中的联合分布

Joint distribution in residue classes of the base-$q$ and Ostrowski digital sums

令$q$为整数$\geq 2$，并且令$S_q(n)$表示基$q$中的$n$的位数之和。对于\[ α=[0;\overline{1,m}],\ m\geq 2, \]，让$S_α(n)$表示$n$的Ostrowski $α$表示中的数字之和。令$m_1,m_2\geq 2$是具有$$\gcd(q-1,m_1)=\gcd(m,m_2)=1.$$的整数我们证明存在$δ>0$，使得对于所有整数$a_1,a_2$，\begin{eqnarray*} &&|\{0\leq n<N: S_{q}(n)\equiv a_1\pmod{m_1},\ S_α(n)\equiv a_2\pmod{m_2}\}| &=&\frac{N}{m_1m_2}+O(N^{1-δ}). \end{eqnarray*}该等式所隐含的渐进关系已由Coquet、Rhin和Toffin证明，并且该等式已由Spiegelhofer证明了$α=[\ \overline{1}\ ]$情况下的等式。

关于不完全剩余域情形的分歧理论

On ramification theory in the imperfect residue field case

本文研究了一类完全离散值域，使得剩余域具有素特征p，且$p$ -基的基数为1。这个类包括二维局部和局部-全局字段。给出了这类域的分歧滤子的一个新定义。看来，一个Hasse-Herbrand型函数可以用所有常见的性质来定义。因此，上分歧群的理论，以及分歧理论的无限扩张，可以开发。接下来，我们考虑一个相等的特征二维局部域$K$。我们在给定域的第二K-群上引入了一些滤子。这种过滤不同于估值引起的过滤。我们证明了二维局部类域论的倒易映射将这种过滤与分歧过滤等同起来。