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Su Hu

Publications and source records attributed to Su Hu.

At least 37 records · Page 2Linked to original sources
arXiv2023

Some new examples of summation of divergent series from the viewpoint of distributions

Let $\{a_{1}, a_{2},\ldots, a_{n},\ldots\}$ be a sequence of complex numbers which has at most polynomial growth and satisfies an extra assumption. In this paper, inspired by a recent work of Sasane, we give an explanation of the sum $$a_{1}+2a_{2}+3a_{3}+\cdots+na_{n}+\cdots,$$ and more generally, for any $k\in\mathbb{N},$ the sum $$1^{k}a_{1}+2^{k}a_{2}+3^{k}a_{3}+\cdots+n^{k}a_{n}+\cdots,$$ from the viewpoint of distributions. As applications, we explain the following summation formulas \begin{equation*} \begin{aligned} 1^{k}-2^{k}+3^{k}-\cdots&=-\frac{E_{k}(0)}{2}, \\ 1^{k}+2^{k}+3^{k}+\cdots&=-\frac{B_{k+1}}{k+1}, \\ ε^{1}1^{k}+ε^{2}2^{k}+ε^{3}3^{k}+\cdots&=-\frac{B_{k+1}(ε)}{k+1}, \end{aligned} \end{equation*} where $E_{k}(0)$, $B_{k}$ and $B_{k}(ε)$ are the Euler polynomials at 0, the Bernoulli numbers and the Apostol--Bernoulli numbers, respectively.

math.NT↗
arXiv2021

On the Stieltjes constants and gamma functions with respect to alternating Hurwitz zeta functions

Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ ζ(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $ζ(z)$, the generalized Stieltjes constants $γ_k(q)$, the Euler constant $γ$, Euler's gamma function $Γ(q)$ and the digamma function $ψ(q)$ have many close connections on their definitions and properties. There are also many integrals, series or infinite product representations of them along the history. In this note, we try to provide a parallel story for the alternating Hurwitz zeta function (also known as the Hurwitz-type Euler zeta function) $$ζ_{E}(z,q)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(n+q)^{z}},$$ the alternating zeta function $ζ_{E}(z)$ (also known as the Dirichlet's eta function $η(z)$), the modified Stieltjes constants $\tildeγ_k(q)$, the modified Euler constant $\tildeγ_{0}$, the modified gamma function $\tildeΓ(q)$ and the modified digamma function $\tildeψ(q)$ (also known as the Nielsen's $β$ function). Many new integrals, series or infinite product representations of these constants and special functions have been found. By the way, we also get two new series expansions of $π:$ \begin{equation*} \frac{π^2}{12}=\frac34-\sum_{k=1}^\infty(ζ_E(2k+2)-1) \end{equation*} and \begin{equation*} \fracπ{2}= \log2+2\sum_{k=1}^\infty\frac{(-1)^k}{k!}\tildeγ_k(1)\sum_{j=0}^kS(k,j)j!. \end{equation*}

math.NT↗
arXiv2021

Mizuno-type result and Wallis' formula

Let $\tildeΓ(z)$ be the modified gamma function introduced by the authors in a recent preprint "arXiv2106.14674". In this note, we obtain the following Mizuno-type result: \begin{equation*} \prod_{m=0}^{\infty}\left\{\prod_{j=1}^{n}(m+z_{j})\right\}^{(-1)^{m}}=\frac{\left(\sqrt{\fracπ{2}}\right)^n}{\prod_{j=1}^{n}\tildeΓ(z_{j})}, \end{equation*} which imply a Kurokawa--Wakayama type formula \begin{equation*} \prod_{m=0}^\infty\left((m+x)^{n}-y^n\right)^{(-1)^{m}} =\frac{\left(\sqrt{\fracπ{2}}\right)^n}{\prod_{ζ^{n}=1}\tildeΓ(x-ζy)} \end{equation*} and a Lerch-type formula \begin{equation*} \prod_{m=0}^\infty(m+x)^{(-1)^{m}}=\frac{\sqrt{\fracπ{2}}}{\tildeΓ(x)}. \end{equation*} By setting $x=1$ in the above result, we recover Wallis' 1656 fomula \begin{equation*}\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot 5}\frac{6\cdot6}{5\cdot 7}\cdots=\fracπ{2}. \end{equation*}

math.NT↗
arXiv2021

Appell-Carlitz numbers

In this paper, we introduce the concept of the (higher order) Appell-Carlitz numbers which unifies the definitions of several special numbers in positive characteristic, such as the Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers.Their generating function is usually named Hurwitz series in the function field arithmetic. By using Hasse-Teichmüller derivatives, we also obtain several properties of the (higher order) Appell-Carlitz numbers, including a recurrence formula, two closed forms expressions, and a determinant expression. The recurrence formula implies Carlitz's recurrence formula for Bernoulli-Carlitz numbers. Two closed from expressions implies the corresponding results for Bernoulli-Carlitz and Cauchy-Carlitz numbers . The determinant expression implies the corresponding results for Bernoulli-Carlitz and Cauchy-Carlitz numbers, which are analogues of the classical determinant expressions of Bernoulli and Cauchy numbers stated in an article by Glaisher in 1875.

math.NT↗
arXiv2021

Generalizations of Lerch's formula by Barnes' multiple zeta functions

The classical Lerch's formula states the following normalized product: $$\prod_{n=0}^\infty(x+n)=\frac{\sqrt{2π}}{Γ(x)},\quad \textrm{Re}(x)>0, $$ where $Γ(x)$ is the Euler gamma function. In this note, by using Barnes' multiple zeta function and its alternating form, we obtain two kinds of generalizations of Lerch's formula, which imply the product $$ \prod_{n=1}^\infty n=\sqrt{2π} $$ (in the sense of zeta regularization) and the product $$\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot 5}\frac{6\cdot6}{5\cdot 7}\cdots=\fracπ{2}$$ (Wallis' formula in 1656), respectively.

math.NT↗
arXiv2021

On the congruences of Eisenstein series with polynomial indexes

In this paper, based on Serre's $p$-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series $G_k$ in the form $$ \sum_{i=1}^n g_i(p)G_{f_i(p)}\equiv g_0(p)\mod p^N, $$ where $f_1(t),\ldots,f_n(t)\in\mathbb{Z}[t]$ are non-constant integer polynomials with positive leading coefficients and $g_0(t),\ldots,g_n(t)\in\mathbb{Q}(t)$ are rational functions. This generalizes the classical von Staudt-Clausen's and Kummer's congruences of Eisenstein series, and also yields some new congruences.

math.NT↗
arXiv2021

Multiple Mertens evaluations

The Mertens' first theorem gives us the following asymptotic formula \begin{equation*} \sum_{\substack{p\leq x\\ p~prime}}\frac{lnp}{p}=lnx+O(1), \end{equation*} and the Mertens' second theorem indicates that there exists a constant $B\approx 0.261$, named the Mertens constant, such that \begin{equation*} \sum_{\substack{p\leq x\\ p~prime}}\frac{1}{p}=ln(lnx)+B+O\left(\frac{1}{lnx}\right). \end{equation*} In this paper, by using the Abel summation formula and Dirichlet's hyperbola method, we extend them to multiple cases.

math.NT↗
arXiv2021

Identities for the Euler polynomials, $p$-adic integrals and Witt's formula

By using Cauchy's formula, it is known that Bernoulli numbers and Euler numbers can be represented by the contour integrals \begin{equation*} \begin{aligned} B_n&=\frac{n!}{2πi}\oint \frac{z}{e^z-1}\frac{d z}{z^{n+1}},\label{condefi}\\[4pt] E_n&=\frac{n!}{2πi}\oint \frac{2e^z}{e^{2z}+1}\frac{d z}{z^{n+1}}, \end{aligned} \end{equation*} while the following Witt's formula represents Euler polynomials through the fermionic $p$-adic integrals $$E_n(a)=\int_{\mathbb Z_p}(x+a)^ndμ_{-1}(x).$$ Base on the above Witt's identity and the binomial theorem, we prove some new identities for the Euler polynomials briefly. In particular, some symmetry properties of Euler polynomials have been discovered, which implies many interesting identities (known or unknown), including the Kaneko-Momiyama type identities (shown by Wu, Sun, and Pan) and the Alzer-Kwong type identity for Euler polynomials.

math.NT↗
arXiv2021

Infinite order linear differential equation satisfied by $p$-adic Hurwitz-type Euler zeta functions

In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $ζ(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $ζ(s)$ satisfies a non-algebraic differential equation and showed that it formally satisfies an infinite order linear differential equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $ζ(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[ζ(s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $ζ(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's differential equation satisfied by $ζ_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ} ζ_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} In contrast with the complex case, due to the non-archimedean property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $ζ_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any finite extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$

math.NT↗
arXiv2020

Cooperative Communications for Internet of Everything in B5G/6G Hybrid and Ubiquitous Networks: Foundation, Further Optimization and Solutions

Cooperative Communications (CC) has been one of most critical communication technologies which plays a founding role on Internet of Everything in B5G/6G networks. As 5G communications standard is gradually established recently, core communications technologies with CC are further studied to significantly improve communication quality and develop new communications scenarios for B5G/6G ubiquitous networks. Considering that CC has been regarded as foundation theory which widely exists in future multiple B5G/6G hybrid scenarios, such as, Cognitive Internet of Things (CIOT) networks, UAVs communications, air-space-ground of integrated networks, underwater acoustic communication and so on, besides it is closely combined with other key technologies, for examples, Massive MIMO, NOMA, Full-duplex transmission, Polar code and so on. Hence, in this paper we review foundation of CC for Internet of Everything in B5G/6G multiple heterogeneous CC networks, and compare fundamental CC algorithms to reveal key of performance improvement. Furthermore we propose that collective communications ideology is theory of foundation to realize communications for arbitrary two points as source/destination devices, sensors, relays, IOT nodes and so on in future.

eess.SP↗
arXiv2020

Hamiltonians for the zeros of a general family of zeta functions

Towards the Hilbert-Pólya conjecture, in this paper, we present a general construction of Hamiltonian $\hat{H}_{f},$ which leads a general family of Hurwitz zeta functions $(-1)^{z_{n}-1}L(f,z_{n},x+1)$ defined by Mellin transform becomes their eigenstates under a suitable boundary condition, and the eigenvalues $E_{n}$ have the property that $z_{n}=\frac{1}{2}(1-iE_{n})$ are the zeros of a general family of zeta functions $L(f,z)\equiv L(f,z,1)$.

math.NT↗
arXiv2020

On $p$-adic Euler $L$-functions

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. Springer, New York, 2007] are also established if we replace the generalized Bernoulli numbers with the generalized Euler numbers.

math.NT↗
arXiv2020

On $p$-adic Hurwitz-type Euler zeta functions

The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it interpolates the Euler polynomials $p$-adically. Many properties are provided for the $p$-adic Hurwitz-type Euler zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the $p$-adic Raabe formula and so on. The definition for the $p$-adic Euler $L$-functions has also been given by using the $p$-adic Hurwitz-type Euler zeta functions.

math.NT↗
arXiv2020

New Complementary Sets with Low PAPR Property under Spectral Null Constraints

Complementary set sequences (CSSs) are useful for dealing with the high peak-to-average power ratio (PAPR) problem in orthogonal frequency division multiplexing (OFDM) systems. In practical OFDM transmission, however, certain sub-carriers maybe reserved and/or prohibited to transmit signals, leading to the so-called \emph{spectral null constraint} (SNC) design problem. For example, the DC sub-carrier is reserved to avoid the offsets in D/A and A/D converter in the LTE systems. While most of the current research focus on the design of low PAPR CSSs to improve the code-rate, few works address the aforementioned SNC in their designs. This motivates us to investigate CSSs with SNC as well as low PAPR property. In this paper, we present systematic constructions of CSSs under SNCs and low PAPR. First, we show that mutually orthogonal complementary sets (MOCSs) can be used as \emph{seed sequences} to generate new CSSs with SNC and low PAPR, and then provide an iterative technique for the construction of MOCSs which can be further used to generate complementary sets (CSs) with low PAPRs and spectral nulls at \emph{varying} positions in the designed sequences. Next, inspired by a recent idea of Chen, we propose a novel construction of these \emph{seed} MOCSs with non-power-of-two lengths from generalized Boolean functions.

cs.IT↗
arXiv2019

Genus theory and Euclidean ideals for real biquadratic fields

In this paper, we use the theory of genus fields to study the Euclidean ideals of certain real biquadratic fields $K.$ Comparing with the previous works, our methods yield a new larger family of real biquadratic fields $K$ having Euclidean ideals; and the conditions for our family seem to be more efficient for the computations. Moreover, the previous approaches mainly focus on the case if $h_K=2$, while the present approach can also deal with the general case when $h_K=2^t (t\geq1)$, where $h_K$ denotes the ideal class number of $K$. In particular, if $h_K\geq 4$, it shows that $H(K)$, the Hilbert class field of $K$, is always non-abelian over $\mathbb{Q}$ for the family of $K$ given in this paper having Euclidean ideals, whereas the previous approaches always requires that $H(K)$ is abelian over $\mathbb{Q}$ explicitly or implicitly. Finally, some open questions have also been listed for further research.

math.NT↗
arXiv2019

Ramanujan expansions of arithmetic functions of several variables over $\mathbb{F}_{q}[T]$

Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in multi-variables over $\mathbb{A}_{+}$ can be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums. These are analogues of classical results over $\mathbb{N}$ by Winter, Delange and Tóth.

math.NT↗