Taekyun Kim
In the recent p-adic q-integral on the p-adic integers' rings was constructed >. The purpose of this paper is to give several interesting integral equation for the p-adic q-integerals on the rings of p-adic integers. As an integral equations, we will treat relations between p-adic q-integral's equations and q-Bernoulli and Euler numbers.
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。
继续探索
We will study p-adic invariant integerals involving trigonometric functions
In [2], I constructed the p-adic q-integral on Zp. In this paper, we consider the properties of the p-adic invariant p-adic q-integral in the ring of p-adic integers at q=-1. Finally we give the some applications of p-adic q-integration at q=-1. These properties are useful and worthwhile to study Euler numbers and polynomials.
$p$-adic multiple zeta values of integer indices
本文研究整数索引的$p$-adic多重ζ值，引入了可接受性和可正规化条件，定义了有限有理线性组合的$p$-adic多重ζ值，并证明了双洗关系成立。
We express the classical $p$-adic integers $\hat{\mathbb{Z}_p}$, as a metric space, as a final colagebra to a certain endofunctor. We realize the addition and the multiplication on $\hat{\mathbb{Z}_p}$ as the coalgebra maps from $\hat{\mathbb{Z}_p} \times \hat{\mathbb{Z}_p}$.
We derive an inequality on the discrepancy of sequences on the ring of $p$-adic integers $\ZZ_p$ using techniques from Fourier analysis. The inequality is used to obtain an upper bound on the discrepancy of the sequence $α_n = na +b$, where $a$ and $b$ are elements of $\ZZ_p$. This is a $p$-adic analogue of the classical LeVeque inequality on the circle group $\RR/\ZZ$.
This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$μ_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,μ_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic pro-unipotent harmonic actions. In this part IV we define and study $p$-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$μ_{N}$'s involve sequences of positive integers ; in this IV-1, by consid…
The explicit formulas of operations, in particular addition and multiplication, of $p $-adic integers are presented. As applications of the results, at first the explicit formulas of operations of Witt vectors with coefficients in $\mathbb{F}_{2}$ are given; then, through solving a problem of Browkin about the transformation between the coefficients of a $p$-adic integer expressed in the ordinary least residue system and the numerically least residue system, similar formulas for Witt vectors with coefficients in $\mathbb{F}_{3}$ are obtained.
In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials {being} strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of $p$-adic integers from polynomials defined over the integers modulo power…
We find upper and lower bounds on the number of rational points that are $ψ$-approximations of some $n$-dimensional $p$-adic integer. Lattice point counting techniques are used to find the upper bound result, and a Pigeon-hole principle style argument is used to find the lower bound result. We use these results to find the Hausdorff dimension for the set of $p$-adic weighted simultaneously approximable points intersected with $p$-adic coordinate hyperplanes. For the lower bound result we show that the set of rational points that $τ$-approximate a $p$-adic integer form a set of resonant points …
We give a sufficient and necessary condition for a p-adic integer to have p-th root in the ring of p-adic integers. The same condition holds clearly for residues modulo p^k. We give a proof that Fermat's last theorem is false for p-adic integers and for residues mod p^k.
We provide some general conditions which ensure that a system of inequalities involving homogeneous polynomials with coefficients in a S-adic field has nontrivial S-integral solutions. The proofs are based on the strong approximation property for Zariski-dense subgroups and adelic geometry of numbers. We give two examples of applications for systems involving quadratic and linear forms.
The essential aim of this paper is to define weighted q-Hardy-littlewood-type maximal operator by means of p-adic q-invariant distribution on Zp. Moreover, we give some interesting properties concerning this type maximal operator.
Some 2-adic integers related to the odd part of 2^e!
本文证明了 OEIS-A359349 中关于 2-adic 整数 z（2^e! 奇数部分的极限）的两个猜想，并利用另一个 2-adic 整数作为关键工具。
We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of $b$-adic integers, $\mathbb Z_{b}, b\in\mathbb N\setminus\{1\}$, by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the `classical' digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of $t,\mathbf T$ and discrepancy.
On $β$-adic expansions of powers of algebraic integer omitting a digit
本文研究了代数整数幂的β进制展开中排除特定数字的性质，证明了其数量随N的变化规律。
Let $f(n)=\sum_k \binom nk^{-1}$. In a previous paper, we defined for a p-adic integer x that f(x) is p-definable if lim $f(x_j)$ exists in $Q_p$, where $x_j$ denotes the mod $p^j$ reduction of $x$. We proved that if p is odd, then -1 is the only element of $Z_p-N$ for which f(x) is p-definable. For p=2, we proved that if the 1's in the binary expansion of x are eventually extraordinarily sparse, then f(x) is 2-definable. Here we present some conjectures that f(x) is 2-definable for many more 2-adic integers. We discuss the extent to which we can prove these conjectures.
For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mat…
The problem of backward dynamics over the ring of p-adic integers is studied. It is shown that Inverse Limit Theory provides the right framework. Backward iterations of a polynomial with p-adic integer coefficients are constructed by solving congruences modulo powers of p, which inturn are solved by Hensel's lifting lemma.
For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $ν_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $ν_{p}(Q(n))$ is shown to be either periodic or unbounded. The first case corresponds to the situation where $Q$ has no roots in the ring of $p$-adic integers. In the periodic situation, the period length is determined.
A linear independence criterion for $p$-adic Hurwitz zeta values at odd integers
该研究推广F. Beukers的定理，建立p进Hurwitz zeta函数在奇数s≥3处取值的线性独立性新准则，通过构造满足f(z)=f(1-z)的Laurent级数的Padé逼近完成证明，补充了p进多伽马值线性独立性的相关工作。
This paper pursues positive characteristic analogues of the results of Furusho, Komori, Matsumoto and Tsumura on $p$-adic multiple $L$-functions. We consider $\infty$-adic and $v$-adic multiple zeta functions concerned by Anglès, Ngo Dac and Tavares Ribeiro. Our main results in this paper consist of: (1) integral expressions of special values of $\infty$-adic multiple zeta functions, (2) integral expression of $v$-adic multiple zeta functions themselves similar to those of $p$-adic multiple $L$-functions, (3) Kummer-type congruence for the special values of $v$-adic multiple zeta functions at …
In this paper are characterized the polynomials, in terms of their coefficients, that have all their orbits dense in the set of 3-adic integers.
Let n be a positive integer and let p be a prime. We calculate the probability that a random monic polynomial of degree n with coefficients in the ring Z_p of p-adic integers splits over Z_p into linear factors.
The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integral on Z_p.
We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give `desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as…
The $3x+1$ map $T$ is defined on the $2$-adic integers $\mathbb{Z}_2$ by $T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. It is still unproved that under iteration of $T$ the trajectory of any rational $2$-adic integer is eventually cyclic. A $2$-adic integer is rational if and only if its representation with $1$'s and $0$'s is eventually periodic. We prove that the $3x+1$ conjugacy $Φ$ maps aperiodic $v\in\mathbb{Z}_2$ onto aperiodic $2$-adic integers provided that $\underline{\lim}\;(\frac{h}{\ell})_{\ell=1}^{\infty} > \frac{\ln(2)}{\ln(3)}$ where $h$ is the number of $1$'s in the fi…
We study the dynamic structures of the monomial $x^m$ over the ring of $p$-adic integers for every positive integer $m$ and for primes $p=2,3$ and $5$. The dynamic structures are described by investigating minimal decompositions which consist of minimal subsystems and attracting basins.
Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Assume that $v_p(f(n))\ge s_1$ and $v_p(g(n))\ge s_2$ hold for all integers $n$ for some $s_1, s_2$ fixed non-negative integers. Let $S$ denote the maximum of $v_p(gcd(f(n),g(n)))$ over all integers $n$. In this paper, we establish multiple lower bound for $v_p(r)$. More specifically, we show that $v_p(r)\ge S-\max{s_1,s_2}+ps_1s_2\frac{p-1}{p-p^{-k}}$, where $k=\lfloor \log_p((p-1)\max\{s_1,s_2\}+1)\rfloor -1$.
The purpose of this paper is to derive the analogue of Lebesgue-Radon-Nikodym theorem with respect to $p$-adic $q$-invariant distribution on $\Bbb Z_p$ which is defined by author in [1].
In this paper, we study some properties of Changhee's q-Bernou lli polynomials which are derived from p-adic invariant integral on Zp. By using these properties, we give some interesting identities related to higher- order q-Bernoulli polynomials.
Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K = Q(zeta) be the p-cyclotomic field and let O_K be the ring of integers of K. Let pi be the prime ideal of K lying over p. An integer B \in O_K is said singular if B^{1/p} not \in K and if B O_K = b^p where b is an ideal of O_K. An integer B \in O_K is said semi-primary if B = beta mod pi^2 where the natural beta is coprime with p. Let sigma be a Q-isomorphism of the field K generating the Galois group Gal(K/Q). When p is irregular, there exists at least one subgroup Gamma of order p of the class grou…
The purpose of this paper is to construct p-adic analytically continued function which interpolates q-Euler numbers at negative integer Finally, we give an explicit p-adic expansion as a power series in n.
Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete system of residues modulo any power of p.
The main purpose of this paper is to prove an identity of symmetry for the Frobenius-Euler polynomials.
Let $p$ be an odd prime number. In this paper, we study the growth of the Sylow $p$-subgroups of the even $K$-groups of rings of integers in a $p$-adic Lie extension. Our results generalize previous results of Coates and Ji-Qin, where they considered the situation of a cyclotomic $\mathbb{Z}_p$-extension. Our method of proof differs from these previous work. Their proof relies on an explicit description of certain Galois group via Kummer theory afforded by the context of a cyclotomic $\mathbb{Z}_p$-extension, whereas our approach is via considering the Iwasawa cohomology groups with coefficien…
This paper deals with a multiple version of zeta- and L-functions both in the complex case and in the p-adic case: [I] Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite linear combination of MZFs with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special…
P-adic approximation of algebraic integers and residue class rings of rings of integer-valued polynomials
本文通过分析Galois扩张中O_K的Q-adic闭包，刻画了使整数值多项式环Int(O_K)模非零素理想的剩余类环满足GE2性质的整数环O_K。
The purpose of this paper is to construct the p-adic twisted (h,q)-Euler-l-function, which interpolates the twisted generalized twisted Euler numbers attached to chi at a negative integer.
Some restricted partition functions in terms of 2-adic valuation
本文研究受限分拆函数的表示问题，利用整数的2进赋值给出了若干此类函数的表达形式。
In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that ${pn \choose n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$.
In this paper, we define the p-adic Euler L-functions using the fermionic p-adic integral on Zp. By computing the values of the p-adic Euler L-functions at negative integers, we show that for Dirichlet characters with odd conductor, this definition is quivalent to the previous definition following Kubata-Leopoldt and Washington's approach. We also study the behavior of p-adic Euler L-functions at positive integers. An interesting thing is that most of the results in Section 11.3.3 of Cohen's book [H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240. S…
In this paper, we study groups of automorphisms of algebraic systems over a set of $p$-adic integers with different sets of arithmetic and coordinate-wise logical operations and congruence relations modulo $p^k,$ $k\ge 1.$ The main result of this paper is the description of groups of automorphisms of $p$-adic integers with one or two arithmetic or coordinate-wise logical operations on $p$-adic integers. To describe groups of automorphisms, we use the apparatus of the $p$-adic analysis and $p$-adic dynamical systems. The motive for the study of groups of automorphism of algebraic systems over $…
For each prime number $p$, the dynamical behavior of the square mapping on the ring $\mathbb{Z}_p$ of $p$-adic integers is studied. For $p=2$, there are only attracting fixed points with their attracting basins. For $p\geq 3$, there are a fixed point $0$ with its attracting basin, finitely many periodic points around which there are countably many minimal components and some balls of radius $1/p$ being attracting basins. All these minimal components are precisely exhibited for different primes $p$.
Let $J$ be an abelian variety over a number field such that the center of its endomorphism ring is equal to the ring of integers. If the endomorphism ring splits at a prime number $l$, then the $l$-adic representation is defined by the minuscule weights (microweights) of simple classical Lie algebras.
We define "splitting functions of level l" for any integer l>0. These functions generalize Dwork's splitting functions : they allow us to represent additive characters of order $p^l$. Then we use these functions to obtain a Stickleberger theorem on the $p$-adic valuation of new Gauss sums.
P-adic Poissonian Pair Correlations via the Monna Map
针对p进整数中具泊松对关联的序列尚无明确实例的问题，本研究提出利用Monna映射将实域中具该性质的序列转换至p进场景的方法，填补了相关空白。
Summation of a large class of the functional series, which terms contain factorials, is considered. We first investigated finite partial sums for integer arguments. These sums have the same values in real and all p-adic cases. The corresponding infinite functional series are divergent in the real case, but they are convergent and have p-adic invariant sums in p-adic cases. We found polynomials which generate all significant ingredients of these series and make connection between their real and p-adic properties. In particular, we found connection of one of our integer sequences with the Bell n…
$p$-adic multiple $L$-functions and twisted multiple Bernoulli numbers
计算Furusho等人引入的$p$-adic多重$L$函数在正整数元组处的特殊值，通过给出循环多重伯努利数与扭曲多重伯努利数关系的显式公式，得到这些特殊值可表示为循环多重调和值的无穷和。
Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding
研究3x+1映射下2-adic整数与奇偶序列的对应，提出逆变换新公式，分析诱导自同构的遍历性、不动点与周期环，并给出具有仿射自相似性的平面嵌入。
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become $\left( m,n\right) $-rings. Second, we introduce a possible $p$-adic analog of the residue class modulo a $p$-adic integer. Then, we find the relations which determine, when the representatives form a $\left( m,n\right) $-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
The 3x+1 function T is defined on the positive integers by $T(x) = \frac{3x+1}{2}$ for x odd and $T(x) = \frac{x}{2}$ for x even. The function T has a natural extension to the 2-adic integers, and there is a continuous function $Φ$ which conjugates T to the 2-adic shift map $σ$. Bernstein and Lagarias conjectured that -1 and 1/3 are the only odd fixed points of $Φ$. In this paper we investigate periodicity associated with $Φ$, a property of the map which is a natural extention of solenoidality. We use it to show that there are nontrivial infinite families of 2-adics that are not fixed points o…
Motivated by a question of Erdős, this paper considers questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers whose expansions use only the digits 0 and 1. The exception set is the set of 3-adic integers whose forward orbits under this action intersects the 3-adic Cantor set infinitely many times. It has been shown that this set has Hausdorff dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these sets leads to study of intersections of multiplicative translates of Can…
We study the problem of simultaneous approximation to a fixed family of real and p-adic numbers by roots of integer polynomials of restricted type. The method that we use for this purpose was developed by H. Davenport and W.M. Schmidt in their study of approximation to real numbers by algebraic integers. This method based on Mahler's Duality requires to study the dual problem of approximation to successive powers of these numbers by rational numbers with the same denominators. Dirichlet's Box Principle provides estimates for such approximations but one can do better. In this thesis we establis…
We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer argument. In this paper, we present rather large class of such p-adic functional series with integer coefficients which contain factorials. By recurrence relations, we constructed sequence of polynomials A_k(n;x) which are a generator for a few other sequences also relevant to some problems in number theory and combinatorics.
Residual Transitivity implies Minimality for Markoff Surfaces over $p$-adic Integers, by Means of $p$-adic Flows
本文用 $p$-adic 流技术给出 Bourgain-Gamburd-Sarnak 定理的替代证明，将模 $p$ 传递性蕴含 $p$-adic 整数点上极小性的结果推广到 $D\equiv 0 \pmod{p^2}$ 或 $(D-4)$ 为非零二次剩余的参数。
We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.
For a set of integers $A$, we consider $R(A)=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values attained by an integral form. This problem has been answered for quadratic forms. Very recently, Antony and Barman have studied this problem for the diagonal binary cubic forms $ax^3+by^3$, where $a$ and $b$ are integers. In this article, we study this problem for diagonal forms. We extend the results of Antony and Barman to the diagonal binary forms $ax^n+by^n$ for all $n\geq 3$. We also study $p$-adic d…
Integer-valued polynomials and $p$-adic Fourier theory
本文针对$p$-进傅里叶理论的开放性问题，利用Katz同构证明$F=\boldsymbol{Q}_{p^2}$时$Λ_F(\frak{X})=o_F[\boldsymbol{[}o_F\boldsymbol{]}]$的等价条件，并通过SageMath计算验证该条件成立的可能性。
The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality…
Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c, p^\infty b + d). When a equiv b mod (p-1) or d \le 0, we express them in terms of p-adic binomial coefficients introduced in a recent paper.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。