Narrow Results

We study rather general multiple zeta functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta functions at non-positive integer points. We first treat the case when the polynomials are power sums, and observe that some “trivial zeros” exist. We also prove that special values are sometimes transcendental. Then we proceed to the general case, and show an explicit expression of special values at non-positive integer points which involves certain period integrals. We give examples of transcendental values of those special values or period integrals. We also mention certain relations among Bernoulli numbers which can be deduced from our explicit formulas. Our proof of explicit formulas are based on the Euler–Maclaurin summation formula, Mahler’s theorem, and a Raabe-type lemma due to Friedman and Pereira.

Let $K$ be a field of characteristic zero and $x$ a free variable. A $K$-${ℰ}$-derivation of $K[x]$ is a $K$-linear map of the form $I,-,\varphi$ for some $K$-algebra endomorphism $\varphi$ of $K[x]$, where $I$ denotes the identity map of $K[x]$. In this paper, we study the image of an ideal of $K[x]$ under some $K$-derivations and $K$-${ℰ}$-derivations of $K[x]$. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ${ℰ}$-derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all $K$-${ℰ}$-derivations and all locally finite $K$-derivations of $K[x]$. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ${ℰ}$-derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent $K$-derivations of $K[x]$, and also for all locally nilpotent $K$-${ℰ}$-derivations of $K[x]$and the ideals $uK[x]$ such that either $u=0$, or $degu\le 1$, or $u$ has at least one repeated root in the algebraic closure of $K$. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified.

In the present paper we propose a new proof of the Grosset–Veselov formula connecting one-soliton solution of the Korteweg–de Vries equation to the Bernoulli numbers. The approach involves Eulerian numbers and Riccati's differential equation.

In this paper, we will study the p-divisibility of multiple harmonic sums (MHS) which are partial sums of multiple zeta value series. In particular, we provide some generalizations of the classical Wolstenholme's Theorem to both homogeneous and non-homogeneous sums. We make a few conjectures at the end of the paper and provide some very convincing evidence.

We define p-adic q-Bernoulli numbers by using a p-adic integral. These numbers have good properties similar to those of the classical Bernoulli numbers. In particular, they satisfy an analogue of the von Staudt–Clausen theorem, which includes information of denominators of p-adic q-Bernoulli numbers.

Zhao established the following harmonic congruence for prime p > 3:

The degenerate Bernoulli numbers β_n(λ) are polynomials with rational coefficients of degree n in the variable λ, which arise in several combinatorial settings. An appropriate change of variable transforms β_n(λ) into a polynomial whose coefficients are all positive. Here, we prove that this transformed polynomial is log-concave, and therefore unimodal. As a consequence, we deduce bounds on the absolute values of the roots of β_n(λ).

For a positive integer n let be the nth harmonic number. In this paper, we prove that for any prime p ≥ 7,

The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis.

Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper, we prove a family of identities involving Bernoulli numbers and apply them to obtain infinitely many weighted sum formulas for double zeta values and triple zeta values where the weight coefficients are given by symmetric polynomials. We give a general conjecture in arbitrary depth at the end of the paper.

In recent years, the congruence

For $k\le n$, let $E(2n,k)$ be the sum of all multiple zeta values with even arguments whose weight is $2n$ and whose depth is $k$. Of course $E(2n,1)$ is the value $\zeta (2n)$ of the Riemann zeta function at $2n$, and it is well known that $E(2n,2)=\frac{3}{4}\zeta (2n)$. Recently Shen and Cai gave formulas for $E(2n,3)$ and $E(2n,4)$ in terms of $\zeta (2n)$ and $\zeta (2)\zeta (2n-2)$. We give two formulas for $E(2n,k)$, both valid for arbitrary $k\le n$, one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers $E(2n,k)$ and for the analogous numbers $E^{\star }(2n,k)$ defined using multiple zeta-star values of even arguments.

This paper is concerned with new results for the circular Eisenstein series ${\epsilon }_r(z)$ as well as with a novel approach to Hilbert–Eisenstein series ${𝔥}_r(z)$, introduced by Michael Hauss in 1995. The latter turns out to be the product of the hyperbolic sinh function with an explicit closed form linear combination of digamma functions. The results, which include differentiability properties and integral representations, are established by independent and different argumentations. Highlights are new results on the Butzer–Flocke–Hauss Omega function, one basis for the study of Hilbert–Eisenstein series, which have been the subject of several recent papers.

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$,

Let $m\ge 1$ be an integer and $p$ be an odd prime. We study sums and lacunary sums of $m$th powers of binomial coefficients from the point of view of arithmetic properties. We develop new congruences and prove the $p$-adic convergence of some subsequences and that in every step we gain at least one or three more $p$-adic digits of the limit if $m=1$ or $m\ge 2$, respectively. These gains are exact under some explicitly given conditions. The main tools are congruential and divisibility properties of the binomial coefficients and multiple and alternating harmonic sums.

Let $m\ge 2$ be an even integer and $p$ be an odd prime. We study Franel-like sums and alternating sums, as well as lacunary sums of $m$th powers of binomial coefficients from the point of view of arithmetic properties. This paper complements the author’s prior work on the cases with $m\ge 1$ odd although, it uses a different approach. It develops new supercongruences and determines the $p$-adic order of these sums as well as of generalized harmonic sums restricted to particular remainder classes modulo $p$.

Wang and Cai established the following harmonic congruence for odd prime and positive integer $r$:

In this paper, we show certain series identities arising from the Jacobi identity of the ordinary theta function. These include several formulas of Ramanujan type given by Berndt and their relevant analogues.

Using Parseval’s identity for the Fourier coefficients of $x^k$, we provide a new proof that $\zeta (2k)=\frac{{(-1)}^{k+1}{B}_{2k}{(2\pi )}^{2k}}{2(2k)!}$.

We prove a general family of congruences for Bernoulli numbers whose index is a polynomial function of a prime, modulo a power of that prime. Our family generalizes many known results, including the von Staudt–Clausen theorem and Kummer’s congruence.