Nicusor Minculete, Diana Savin
In this paper, we generalize the arithmetic functions $τ$ and $τ^{\left(e\right)}$ in algebraic number fields and we find some properties of these functions.
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Let $\left( g\left( x \right),xg\left( x \right) \right)$ be a Riordan matrix from the Bell subgroup. We denote ${{\left( g\left( x \right),xg\left( x \right) \right)}^{φ}}=\left( {{g}^{\left( φ\right)}}\left( x \right),x{{g}^{\left( φ\right)}}\left( x \right) \right)$, where a matrix power is defined in the standard way. The polynomials ${{c}_{n}}\left( x \right)$ such that ${{g}^{\left( φ\right)}}\left( x \right)=\sum\nolimits_{n=0}^{\infty }{{{c}_{n}}}\left( φ\right){{x}^{n}}$ will be called composition polynomials. We consider the composition polynomials of the RNA matrix. The construction…
We consider the group of the matrices $\left( 1,g\left( x \right) \right)$ isomorphic to the group of formal power series $g\left( x \right)=x+{{g}_{2}}{{x}^{2}}+...$ under composition: $\left( 1,{{g}_{2}}\left( x \right) \right)\left( 1,{{g}_{1}}\left( x \right) \right)=\left( 1,{{g}_{1}}\left( {{g}_{2}}\left( x \right) \right) \right)$. Denote $P_{k}^{α}=\left( 1,x{{\left( 1-kα{{x}^{k}} \right)}^{{-1}/{k}\;}} \right)$. Matrix $\left( 1,g\left( x \right) \right)$is decomposed into an infinite product of the matrices $P_{k}^{α}$ with suitable exponents in two ways: to left-handed and right-han…
Relatively Prime Sets, Divisor Sums, and Partial Sums
对于正整数的非空有限集$A$，设$\gcd\left(A\right)$表示$A$中元素的最大公约数。令$f\left(n\right)$和$Φ\left(n\right)$分别表示使得$\gcd\left(A\right) = 1$的$\left\{1, 2, \ldots, n\right\}$的子集$A$的数目和使得$\gcd\left(A\cup\left\{n\right\}\right) =1$的$\left\{1, 2, \ldots, n\right\}$的子集$A$的数目。令$D\left(n\right)$是$f\left(n\right)$的除数和。本文得到了$f\left(n\right)$，$Φ\left(n\right)$和$D\left(n\right)$的部分和.我们还得到了$D\left(n\right)$的组合解释和同余性质。在本文的最后，我们给出了有关$Φ\left(n\right)$和$D\left(n\right)$的开放性问题。
Each numerical sequence $\left( {{b}_{0}},{{b}_{1}},{{b}_{2}},... \right)$ with the generating function $B\left( x \right)$ defines the pseudo-involution in the Riordan group $\left( 1,xg\left( x \right) \right)$ such that $g\left( x \right)=1+xg\left( x \right)B\left( {{x}^{2}}g\left( x \right) \right)$. In the present paper we realize a simple idea: express the coefficients of the series ${{g}^{m}}\left( x \right)$ in terms of the coefficients of the series $B\left( x \right)$. Obtained expansion has a bright combinatorial character, sheds light on the connection of the pseudo-involution in …
Let $\left(a_{n}\right)_{n}$ be a strictly increasing sequence of positive integers, denote by $A_{N}=\left\{ a_{n}:\,n\leq N\right\} $ its truncations, and let $α\in\left[0,1\right]$. We prove that if the additive energy $E\left(A_{N}\right)$ of $A_{N}$ is in $Ω\left(N^{3}\right)$, then the sequence $\left(\left\langle αa_{n}\right\rangle \right)_{n}$ of fractional parts of $αa_{n}$ does not have Poissonian pair correlations (PPC) for almost every $α$ in the sense of Lebesgue measure. Conversely, it is known that $E\left(A_{N}\right)=\mathcal{O}\left(N^{3-\varepsilon}\right)$, for some fixed …
Let $Λ\left(n\right)$ be the von Mangoldt function and $r_{G}\left(n\right) := \sum_{m_1 + m_2=n} Λ\left(m_1 \right) Λ\left(m_2 \right)$ be the counting function for the numbers that can be written as sum of two primes (that we will call "Goldbach numbers", for brevity) and let $\widetilde{S }\left(z\right) := \sum_{n\geq1} Λ\left(n\right) e^{-nz}$, with $z\in\mathbb{C}$, $\mathrm{Re}\left(z\right)>0$. In this paper we will prove the identity $$\widetilde{S}\left(z\right) = \frac{e^{-2z}}{z}-\sum_ρz^{-ρ} Γ\left(ρ\right) + \sum_ρ \left(z^{-ρ} γ\left(ρ,2z\right) - \frac{2^ρe^{-z}}ρ \right) + G\l…
Lower Bounds for Densities of Transcendental Gamma-Function Derivatives
研究超越函数导数在正整点和有理移位整点上的超越性，推导其密度下界。
Identities involving the $\left(h,q\right)$-Genocchi polynomials and $\left(h,q\right)$-Zeta-type function
本文的主要目的是利用$q$ -伯恩斯坦多项式中提到的$\mathbb{Z}_{p}$上的费米子$p$ -adic $q$ -积分，得到$\left(h,q\right)$ -Genocchi数和多项式的一些有趣的性质.本文考虑T. Kim提出的一种新的生成函数，它也可以通过对$\left(h,q\right)$ -Genocchi多项式的生成函数进行Mellin变换而得到，本文研究了$\left(h,q\right)$ -Zeta型函数。本文导出了$\left(h,q\right)$ -Zeta函数的对称性质，并由这些性质给出了$\left(h,q\right)$ -Genocchi多项式的对称性质。
Let $Λ\left(n\right)$ be the von Mangoldt function and $r_{SP}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}Λ\left(m_{1}\right)Λ\left(m_{2}\right)Λ\left(m_{3}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares of primes. Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_{i}\left(N,k\right),\, i=1,\dots,4$ suitable parameters depending on $Γ(s)$. We prove that $$\sum_{n\leq N}r_{SP}\left(n\right)\frac{\left(N-n\right)^{k}}{Γ\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O…
"Infinitely Often" Transcendence of Gamma-Function Derivatives
研究伽马函数在任意非负整数点以外的有理数点处导数的超越性，证明对于特定q，导数序列中存在无限多个超越数，并给出超越数密度的下界。
Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. Generalized Euler polynomials ${{g}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( m \right)$ is the polynomial of degree $\le n$, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan matrices. Generalized Narayana polynomials ${{h}_{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits_{m=0}^{\infty }{\left( m+1 \right)..…
Hausdorff dimension of some exceptional sets in Lüroth expansions
本文研究Lüroth展开中数字增长率的度量理论，完全确定了与数字增长率极限相关的三类例外集的Hausdorff维数。
The diophantine equation $\left(2^{k}-1\right)\left(3^{k}-1\right)=x^{n}$
本文考察特定指数型丢番图方程，证明其在 $k、x$ 为正整数且 $n>2$ 时无解。
Let $Λ$ be the von Mangoldt function and $r_{Q}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}Λ\left(m_{1}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares (that we will call Linnik numbers, for brevity). Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_{i}\left(N,k\right),\, i=1,\dots,k$ suitable parameters depending on $J_{v}\left(u\right)$, where $J_{v}\left(u\right)$ denotes the Bessel function of complex order $v$ and real argument $u$. We prove that $$ \sum_{n\leq N}r_{Q}\left(n\right)\frac{\left(N-n\right)^{k}}{Γ\left(k…
Two theorems are demonstrated giving analytical expressions of the fundamental solutions of the Pell equation $X^{2}-DY^{2}=1$ found by the method of continued fractions for two squarefree polynomial expressions of radicands of Richaud-Degert type $D$ of the form $D=\left(f\left(u\right)\right)^{2}\pm2^αn$, where $D$, $n>0$, $α\geq0,\in\mathbb{Z}$, and $f\left(u\right)>0,\in\mathbb{Z}$, any polynomial function of $u\in\mathbb{Z}$ such that $f\left(u\right)\equiv0\left(mod\,\left(2^{α-1}n\right)\right)$ or $f\left(u\right)\equiv\left(2^{α-2}n\right)\left(mod\,\left(2^{α-1}n\right)\right)$.
Let $Λ\left(n\right)$ be the Von Mangoldt function, let \[ r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}Λ\left(m_{1}\right)Λ\left(m_{2}\right), \] \[ r_{PT}\left(N,h\right)=\sum_{n=0}^{N}Λ\left(n\right)Λ\left(n+h\right),\,h\in\mathbb{N} \] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the averages of $r_{G}\left(n\right)$ and $r_{PT}\left(N,h\right)$ in terms of elementary functions, the incomplete Beta function $B_{z}\left(a,b\…
Generalized Euler polynomials ${{α}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( x \right)$ is the polynomial of degree $n$, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan arrays. Generalized Narayana polynomials ${{φ}_{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits_{m=0}^{\infty }{\left( m+1 \right)...\left( m+n \right){{p}_{n}}}\left( m \right){{x}^{m}}$ are the numerator polynomials of the generating functions of diagonals of the expo…
In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method.
An explicit decomposition formula of a matrix in $GL_{2}\left(\mathbb{Z}\right)$
本文针对$GL_{2}\left(\mathbb{Z}\right)$矩阵分解问题，提出了一种显式公式，可将任意矩阵$M$分解为由特定元素$A$、$B$和$C$生成的子群中。
We show the following bounds on the prime counting function $π(x)$ using principles from analytic number theory, giving an estimate: $$2 \log 2 \geq \limsup_{x \rightarrow \infty} \frac{π(x)}{x / \log x} \geq \liminf_{x \rightarrow \infty} \frac{π(x)}{x / \log x} \geq \log 2$$ for all $x$ sufficiently large. We also conjecture about the bounding of $π((x+1)^{2}) - π(x^{2})$, as is relevant to Legendre's conjecture about the number of primes in the aforementioned interval such that: $$ \left \lfloor\frac{1}{2}\left(\frac{\left(x+1\right)^{2}}{\log\left(x+1\right)}-\frac{x^{2}}{\log x}\right)-\f…
Stieltjes constants appearing in the Laurent expansion of the hyperharmonic zeta function
本文研究了超调和Zeta函数的亚纯延拓，确立了其Laurent展开中自然出现的常数，并证明了这些常数与广义指数积分的积分均可表示为特殊常数的有限组合。
Let $S=\left\langle s_1,\ldots,s_n\right\rangle$ be a numerical semigroup generated by the relatively prime positive integers $s_1,\ldots,s_n$. Let $k\geqslant 2$ be an integer. In this paper, we consider the following $k$-power variant of the Frobenius number of $S$ defined as $${}^{k\!}r\!\left(S\right):= \text{ the largest } k \text{-power integer not belonging to } S.$$In this paper, we investigate the case $k=2$. We give an upper bound for ${}^{2\!}r\!\left(S_A\right)$ for an infinite family of semigroups $S_A$ generated by {\em arithmetic progressions}. The latter turns out to be the exa…
We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $ζ(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value $t\left(2,\ldots,2,1\right)$.
In this paper, we use Abel's summation formula to evaluate several quadratic and cubic sums of the form: \[{F_N}\left( {A,B;x} \right) := \sum\limits_{n = 1}^N {\left( {A - {A_n}} \right)\left( {B - {B_n}} \right){x^n}} ,\;x \in [ - 1,1]\] and \[F\left( {A,B,ζ(r)} \right): = \sum\limits_{n = 1}^\infty {\left( {A - {A_n}} \right)\left( {B - {B_n}} \right)\left( {ζ\left( r \right) - {ζ_n}\left( r \right)} \right)} ,\] where the sequences $A_n,B_n$ are defined by the finite sums ${A_n} := \sum\limits_{k = 1}^n {a_k} ,\ {B_n} := \sum\limits_{k = 1}^n {b_k}\ ( {a_k},{b_k} =o(n^{-p}),{\mathop{\Re}\n…
On the Gan-Gross-Prasad conjecture and its refinement for $\left(\mathrm{U}\left(2n\right),\mathrm{U}\left(1\right)\right)$
本文一般地证明了酉群对 (U(2n),U(1)) 的 Gan-Gross-Prasad 猜想，并在一定假设下建立中心 L 值的 Ichino-Ikeda 型精细公式；同时证明其分裂类比 (GL_{2n},GL_1) 的相应猜想与精细形式。
On the Gross-Prasad conjecture with its refinement for $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the generalized Böcherer conjecture
本文通过组合多个theta对应，在（SO(5), SO(2)）情形下证明了Bessel周期的Ichino-Ikeda型公式，并由此将Böcherer猜想推广到非平凡环面特征情形。
Given $n\in\mathbb{N}$, let $ω\left(n\right)$ denote the number of distinct prime factors of $n$, let $Z$ denote a standard normal variable, and let $P_{n}$ denote the uniform distribution on $\left\{ 1,\ldots,n\right\} $. The Erdős-Kac Theorem states that $$P_{n}\left(m\le n:ω\left(m\right)-\log\log n\le x\left(\log\log n\right)^{1/2}\right)\to\mathbb{P}\left(Z\le x\right)$$ as $n\to\infty$; i.e., if $N\left(n\right)$ is a uniformly distributed variable on $\lbrace 1,\ldots,n \rbrace$, then $ω\left(N\left(n\right)\right)$ is asymptotically normally distributed as $n\to \infty$ with both mean …
An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} α\right)_{n\geq 1}$ is uniformly distributed modulo one for almost all $α$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $\left(\left\{a_{n} α\right\}\right)_{n \geq 1}$ for almost all $α$. In particular it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $\left(a_{n}\righ…
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