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Min-Soo Kim

Publications and source records attributed to Min-Soo Kim.

At least 55 records · Page 3Linked to original sources
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.

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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.

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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 $p-1.$

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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 $L(f,z)\equiv L(f,z,1)$.

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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.

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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.

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arXiv2020

On Alzer-Kwong's Identities for Bernoulli polynomials

In this paper, we prove new identities for Bernoulli polynomials that extend Alzer and Kwong's results. The key idea is to use the Volkenborn integral over $\mathbb Z_p$ of the Bernoulli polynomials to establish recurrence relations on the integrands. Also, some known identities are obtained by our approach.

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arXiv2019

An analogue of Kummer's relation between the ideal class number and the unit index of cyclotomic fields

In this paper, we obtain a formula for the special value of Euler-Dirichlet $L$-function $L_E(s,χ)$ at $s=1$. This leads to another class number formula of $\mathbb{Q}(μ_{m})^{+}$, the maximal real subfield of $m$th cyclotomic field. From this formula, we construct a new type of cyclotomic units in $\mathbb{Q}(μ_{p^{n}})$, which implies a similar Kummer's relation between the ideal class number of $\mathbb{Q}(μ_{p^{n}})^{+}$ and the unit index.

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arXiv2019

On Dirichlet's lambda functions

Let $$λ(s)=\sum_{n=0}^\infty\frac1{(2n+1)^s},$$ $$β(s)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(2n+1)^s},$$ and $$η(s)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}$$ be the Dirichlet lambda function, its alternating form, and the Dirichlet eta function, respectively. According to a recent historical book by Varadarajan (\cite[p.~70]{Varadarajan}), these three functions were investigated by Euler under the notations $N(s)$, $L(s)$, and $M(s)$, respectively. In this paper, we shall present some additional properties for them. That is, we obtain a number of infinite families of linear recurrence relations for $λ(s)$ at positive even integer arguments $λ(2m)$, convolution identities for special values of $λ(s)$ at even arguments and special values of $β(s)$ at odd arguments, and a power series expansion for the alternating Hurwitz zeta function $J(s,a)$, which involves a known one for $η(s)$.

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arXiv2019

Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture

In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\ldots, G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely many G-irregular primes and establish non-trivial lower bounds for their number up to a given bound $x$ as $x$ tends to infinity. As a by-product, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-primitive root.

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arXiv2018

On the irregular primes with respect to Euler polynomials

An odd prime $p$ is called irregular with respect to Euler polynomials if it divides the numerator of one of the numbers $$E_1(0),E_{3}(0),\ldots,E_{p-2}(0),$$ where $E_n(x)$ is the $n$-th Euler polynomial. As in the classical case, we link the regularity of primes to the divisibility of some class numbers. Besides, we obtain some results on the distribution of such irregular primes. Remark: This preprint has been withdrawn, because all the results in it have been included or improved in a recent preprint. (See arXiv:1809.08431).

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arXiv2018

On $p$-adic multiple Barnes-Euler zeta functions and the corresponding log gamma functions

Suppose that $ω_1,\ldots,ω_N$ are positive real numbers and $x$ is a complex number with positive real part. The multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ with parameter vector $\barω=(ω_1,\ldots,ω_N)$ is defined as a deformation of the Barnes multiple zeta function as follows $$ ζ_{E,N}(s,x;\barω)=\sum_{t_1=0}^\infty\cdots\sum_{t_N=0}^\infty \frac{(-1)^{t_1+\cdots+t_N}}{(x+ω_1t_1+\cdots+ω_Nt_N)^s}. $$ In this paper, based on the fermionic $p$-adic integral, we define the $p$-adic analogue of multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ which we denote by $ζ_{p,E,N}(s,x;\barω).$ We prove several properties of $ζ_{p,E,N}(s,x; \barω)$, including the convergent Laurent series expansion, the distribution formula, the difference equation, the reflection functional equation and the derivative formula. By computing the values of this kind of $p$-adic zeta function at nonpositive integers, we show that it interpolates the higher order Euler polynomials $E_{N,n}(x;\barω)$ $p$-adically. Furthermore, we define the corresponding multiple $p$-adic Diamond-Euler Log Gamma function. We also show that the multiple $p$-adic Diamond-Euler Log Gamma function ${\rm Log}\, Γ_{\! D,E,N}(x;\barω)$ has an integral representation by the multiple fermionic $p$-adic integral, and it satisfies the distribution formula, the difference equation, the reflection functional equation, the derivative formula and also the Stirling's series expansions.

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arXiv2018

Jackson's integral of multiple Hurwitz-Lerch zeta functions and multiple gamma functions

Using the Jackson integral, we obtain the $q$-integral analogue of the Raabe type formulas for Barnes multiple Hurwitz-Lerch zeta functions and Barnes and Vardi's multiple gamma functions. Our results generalize $q$-integral analogue of the Raabe type formulas for the Hurwitz zeta functions and log gamma functions in [N. Kurokawa, K. Mimachi, and M. Wakayama, Jackson's integral of the Hurwitz zeta function, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 1, 43--56]. During the proof we also obtain a new formula on the relationship between the higher and lower orders Hurwitz zeta functions.

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arXiv2017

Special values and integral representations for the Hurwitz-type Euler zeta functions

The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} ζ_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $ζ_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.

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