Jianqiang Zhao
In this paper we will study the p-divisibility of partial sums of multiple zeta value series. In particular we provide some generalizations of the classical Wolstenholme's Theorem.
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We give a generalization of Wolstenholme's harmonic series congruence for the Lucas sequences.
Multiple harmonic sums are iterated generalizations of harmonic sums. Recently Dilcher has considered congruences involving q-analogs of these sums in depth one. In this paper we shall study the homogeneous case for arbitrary depth by using generating functions and shuffle relations of the q-analog of multiple harmonic sums. At the end, we also consider some non-homogeneous cases.
We establish a q-analogue of Wolstenholme's harmonic series congruence.
$p$-adic valuation of harmonic sums and their connections with Wolstenholme primes
本研究围绕Eswarathasan-Levine关于调和数$p$进赋值的猜想展开，用模算术方法分别证明了非Wolstenholme素数与Wolstenholme素数情形下，$H(p^mn)$的$p$进赋值在$H(n)$的$p$进赋值为3时的反常渐近行为。
We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$) involves a linear combination of $n$ multiple harmonic sums, and holds $\mod{p^{2n+3}}$. The coefficients in these congruences are integers depending on $n$ and $k$, but independent of $p$. More generally, we construct a family of congruences for ${kp-1\choose p-1} \mod{p^{2n+3}}$, whose members contain a variable number of terms, and show that in this family …
In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$ 1+\frac 12 +\frac 13+...+\frac{1}{p-1} $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$ 1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2} $$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. Namely…
A proof of Wolstenholme's theorem and congruence properties via an Egorychev-type integral
本文通过埃戈里切夫型积分和指数变量变换，详细证明了沃尔斯滕霍尔姆定理，并明确展示了与调和和与伯努利数的联系，推导了模p^4的经典细化，并精确提取了B_{p-3}项。
\medskip\noindent\textbf{Résumé.} Soit $l$ un entier et $\ors=(s_1, \dots, s_l)$ une séquence d'entiers positifs. Dans ce document, nous étudierons les propriétés arithmétique de sommes harmoniques multiples $H(\ors; n)$, qui est le $n$-ème somme partielle de la valeur de la série multiple zeta $ζ(\ors)$. On conjecture que pour tout $\ors$ et de tous les premiers $p$, il n'y a que de nombreux finitely $p$-partie intégrante sommes $H(\ors,n)$. Ceci généralise une conjecture de Eswarathasan et Levine et Boyd pour la série harmonique. Nous fournissons beaucoup d'éléments de preuve pour cette conj…
Recently, Bradley studied partial sums of multiple q-zeta values and proved a duality result. In this paper, we present a generalization of his result.
The duality is a fundamental property of the finite multiple harmonic sums (MHS). In this paper, we prove a duality result for certain generalizations of MHS which appear naturally as the differences of MHS. We also prove a formula for the differences of these generalized MHS.
Iterated harmonic numbers
本文提出迭代调和数这一调和数的自然推广，证明其保留调和数的若干经典性质并推广欧拉常数，同时猜想所有迭代调和数中仅第一个为整数，并给出 j=2、3 的数值证据。
In Monstrous moonshine, genus 0 property and the notion of replicability are strongly connected. With regards to recent developments of moonshine, we investigate a higher genus generalization of replicability for a general automorphic form. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera to obtain number theoretic generalizations of important results in Monstrous moonshine. Furthermore, we show the utility of the extended notions in yielding uniform proofs for numerous arithmetic properties of Fourier coe…
We calculate the special values of the spectral zeta function of the non-commutative harmonic oscillator, and give a general formula for them as integrals of certain algebraic functions. This is a generalization of the result by Ichinose-Wakayama (2005), in which the first two special values are studied.
The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(τ)$ in terms of the Hecke system of $\operatorname{SL}_2(\mathbb{Z})$-modular functions $j_n(τ)$. This formula can be reformulated entirely number-theoretically. Namely, it is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form in $τ$ with a pole at $z$. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves.…
In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations are useful to obtain closed forms of some series related to harmonic numbers.
We introduce adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values. They are two variants of cyclotomic multiple zeta values, closely related to each other. They arise as key tools for the study of $p$-adic cyclotomic multiple zeta values. Moreover, cyclotomic multiple harmonic values provide an adelic lift to a cyclotomic generalization of finite multiple zeta values. We establish certain standard properties of these two objects. We consider two types of properties : some related to double shuffle relations, and some related to associator and Kashiwara-Vergne relatio…
A generalization of the Apery-like numbers, which is used to describe the special values $ζ_Q(2)$ and $ζ_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a recurrence relation for them, which shows a ladder structure among them. Further, we consider the `rational part' of the higher Apery-like numbers. We discuss several kinds of congruence relations among them, which are regarded as an analogue of the ones among Apery numbers.
Generalized Harmonic Progression
本文提出了一种广义调和级数的求和公式，结合Faulhaber公式推导出新的傅里叶级数部分和公式，扩展了前人研究成果。
Kempner-like harmonic series
本文重新审视了数字受限的 Kempner-like 调和级数，证明了二进制数字和相关的极限为 2 log 2，并将其推广至任意 b 进制数字和与块计数函数，分别求得相应极限值。
Applications of a formula of Maesaka-Seki-Watanabe type for multiple harmonic $q$-sums
本文利用 Maesaka-Seki-Watanabe 型公式的 $q$-模拟，给出了多重 zeta 值的 $q$-模拟的对偶性的另一种证明，并证明了 Kawashima 函数的 $q$-模拟的一个恒等式。
In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. It is shown that harmonic versions of these polynomials and their generalizations are useful to obtain closed forms of some series related to harmonic numbers.
We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.
We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random walks on a finite number of sites. Relations among multiplicative nested sums, which are generalizations of relations between harmonic series and multiple zeta functions, can be easily derived from identities of the index matrices. Combinatorial identities and their generalizations can also be derived from this computation.
Maesaka-Seki-Watanabe's formula for multiple harmonic $q$-sums
本文提出了MSW公式的$q$-模拟，利用该模拟证明了本原单位根处有限多重调和$q$-级数的对偶关系，并给出了Yamamoto对Schur型MSW公式推广的$q$-模拟。
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