On the number of pairs of positive integers x1, x2 <= H such that x1 x2 is a k-th power

关于正整系数Laurent多项式的哥德巴赫定理

A Goldbach theorem for Laurent polynomials with positive integer coefficients

本文证明了正整系数Laurent多项式情形下的哥德巴赫猜想类比定理。

Positive density of integer polynomials with some prescribed properties

In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer $n$ and any set of $n$ distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree $n$, height at most $H$ and Galois group $S_n$ whose roots are close to the given $n$ points.

On certain positive integer sequences

A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.

Special values of generalized multiple Hurwitz zeta function at non-positive integers

In this paper, we provide an alternative method to calculate the values of generalized multiple Hurwitz zeta function at non-positive integers by means of \emph{Raabe}'s formula and the \textit{Bernoulli} numbers.

Analytic continuation of multiple zeta-functions and the asymptotic behavior at non-positive integers

We prove two theorems. Theorem 1 gives the meromorphic continuation of the multiple zeta function to the whole space. In Theorem 2, we prove asymptotic behavior near the non-positive integers.

On sequences of positive integers having no $p$ terms in arithmetic progression

We use topological ideas to show that, assuming the conjecture of Erd\"(o)s on subsets of positive integers having no $p$ terms in arithmetic progression (A. P.), there must exist a subset $M_p$ of positive integers with no $p$ terms in A. P. with the property that among all such subsets, $M_p$ maximizes the sum of the reciprocals of its elements.

Representing positive integers as a sum of a squarefree number and a small prime

We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.

偶数正整数的2-复杂度

The 2-complexity of even positive integers

本文提出l-复杂度概念，研究偶数正整数的2-复杂度，证明基本结果并提出相关问题。

On Diophantine approximations with positive integers: a remark to W.M.Schmidt's theorem

We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $α_1x_1+α_2x_2 +y$ with {\it positive} integers $x_1,x_2$.

Potential energy of totally positive algebraic integers

Given positive real numbers, we prove two inequalities involving their potential energy and their power sums. We also prove an inequality involving the energy and the discriminant and apply it to deduce a result on totally positive irreducible polynomials.

Polynomials Consisting of Quadratic Factors with Roots Modulo Any Positive Integer

We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.

Instant Multiple Zeta Values at Non-Positive Integers and Bernoulli Functions

We give an instant evaluation of multiple Zeta function at non-positive integers by elementary methods and discuss the Fourier theory (on unit interval) of the product of Bernoulli polynomials.We also show that the polynomial expression for Hurwitz Zeta function(at non-positive integral values of the first variable) and the polynomial expression for Bernoulli polynomials are equivalent.

Is there an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the number of integer solutions, if the solution set is finite?

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let f(n) denote the greatest finite total number of solutions of a subsystem of E_n in integers x_1,...,x_n. We prove: (1) the function f is strictly increasing, (2) if a non-decreasing function g from positive integers to positive integers satisfies f(n) \geq g(n) for any n, then a finite-fold Diophantine representation of g does not exist, (3) if the question of the title has a positive answer, then there is a computable strictly increasing function g from positive integers to positive integers su…

On Frobenius problem with restrictions on common divisors of coefficients

Let $m,s,t$ are positive integers with $t\leq s-2$ and $a_1,a_2,\ldots,a_s$ are positive integers such that $(a_1,a_2,\ldots,a_{s-1})=1$. In the paper we prove that every sufficiently large positive integer can be written in the form $a_1μ_1+a_2μ_2+\ldots+a_sμ_m$, where positive integers $μ_1,μ_2,\ldots,μ_s$ have no common divisor being $m$-th power of a positive integer greater than $1$ but each $t$ of the values of $μ_1,μ_2,\ldots,μ_n$ have a common divisor being $m$-th power of a positive integer greater than $1$. Moreover, we show that every sufficiently large positive integer can be writt…

Values of zeta-one functions at positive even integers

Motivated by Euler-Goldbach and Shallit-Zikan theorems, we introduce zeta-one functions with infinite sums of $n^{s}\pm1$ as an analogy of the Riemann zeta function. Then we compute values of these functions at positive even integers by the residue theorem.

关于Dirichlet $L$-函数在正整数处乘积的均值

On the mean values of products of Dirichlet $L$-functions at positive integers

本文研究Dirichlet $L$-函数在正整数处乘积的均值分布，给出了由Dirichlet特征加权、涉及Bernoulli函数和Jordan totient函数的显式公式，推广了多种已知结果。

Representations of integers by certain positive definite binary quadratic forms

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.

非正整数处的斯特林多项式与多重 zeta（星）函数

Stirling polynomials and multiple zeta (star) functions at non-positive integers

本文利用斯特林多项式给出多重 zeta 函数及其星函数在非正整数处取值的显式公式，并揭示二者之间以及反向值与广义格雷戈里系数之间的联系。

Quadratic forms representing all odd positive integers

We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19 of those represent all positive odds. (Jagy later dealt with a 20th candidate.) Assuming that the remaining three forms represent all positive odds, we prove that an arbitrary, positive-definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451. This result is analogous to Bhargava and Hanke's celebrated…

Fourier Analysis and the closed form for the Zeta Function at even positive integers

Using a summation identity obtained for the Fourier coefficients of $x^{2k}$, we derive a closed form expression for the zeta function at even positive integers, using a technique similar to one in an existing proof by Aladdi and Defant[1], but in a simpler and shorter way.

The representation of integers by positive ternary quadratic polynomials

An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form $Q({\mathbf x} + {\mathbf v})$, where $Q$ is an integral quadratic form in the variables ${\mathbf x} = (x_1, \ldots, x_n)$ and ${\mathbf v}$ is a vector in ${\mathbb Q}^n$. Its conductor is defined to be the smallest positive integer $c$ such that $c{\mathbf v} \in {\mathbb Z}^n$. We prove that for a fixed positive integer $c$, there are only finitely many equ…

Revisiting The Riemann Zeta Function at Positive Even Integers

Using Parseval's identity for the Fourier coefficients of $x^k$, we provide a new proof that $ζ(2k)=\dfrac{(-1)^{k+1}B_{2k}(2π)^{2k}}{2(2k)!}$.

Additive structure of totally positive quadratic integers

Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and determine its generators (indecomposable integers) and relations; they can be nicely described in terms of the periodic continued fraction for $\sqrt D$. We also characterize all uniquely decomposable integers in $K$ and estimate their norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$ completely determines the real quadratic field $K$.

Positive alternating sums of integer partitions

We combine an extended version of Bailey's transform with an identity of Bressoud and with some identities of Berkovich and Warnaar to prove a variety of positivity results for alternating sums involving partition functions.

Some new results on consecutive equidivisible integers

We have found maximum possible runs of consecutive positive integers each having exactly $k$ divisors for some fixed values of $k$. In addition, we exhibit the run of 10 consecutive positive integers each having exactly 12 divisors and two runs of 14 consecutive positive integers each having exactly 24 divisors.

Values at non-positive integers of partially twisted multiple zeta-functions I

We study the behavior of partially twisted multiple zeta-functions. We give new closed and explicit formulas for special values at non-positive integer points of such zeta-functions. Our method is based on a result of M. de Crisenoy on the fully twisted case and the Mellin-Barnes integral formula.

Counting $r$-tuples of positive integers with $k$-wise relatively prime components

Let $r\ge k\ge 2$ be fixed positive integers. Let $\varrho_{r,k}$ denote the characteristic function of the set of $r$-tuples of positive integers with $k$-wise relatively prime components, that is any $k$ of them are relatively prime. We use the convolution method to establish an asymptotic formula for the sum $\sum_{n_1,\ldots,n_r\le x} \varrho_{r,k}(n_1,\ldots,n_r)$ by elementary arguments. Our result improves the error term obtained by J. Hu (2013).

"Choix de Bruxelles": A New Operation on Positive Integers

The "Choix de Bruxelles" operation replaces a positive integer n by any of the numbers that can be obtained by halving or doubling a substring of the decimal representation of n. For example, 16 can become any of 16, 26, 13, 112, 8, or 32. We investigate the properties of this interesting operation and its iterates.

Which numbers are not the sum plus the product of three positive integers?

We investigate the number $R_3(n)$ of representations of $n$ as the sum plus the product of three positive integers. On average, $R_3(n)$ is $\frac{1}{2}\log^2 n$. We give an upper bound for $R_3(n)$ and an upper bound for the number of $n \leq N$ such that $R_3(n) = 0$. We conjecture that $R_3(n)= 0$ infinitely often.

On the number of pairs of positive integers $x, y \le H$ such that $x^2 + y^2 + 1$ is squarefree

It is not difficult to find an asymptotic formula for the number of pairs of positive integers $x, y \le H$ such that $x^2 + y^2 + 1$ is squarefree. In the present paper we improve the estimate for the error term in this formula using the properties of certain exponential sums. A.Weils's estimate for the Kloosterman sum plays the major role in our analysis.

Representation of positive integers by the form $x^3+y^3+z^3-3xyz$

We study the number $ν(n)$ of representations of a positive integer $n$ by the form $x^3+y^3+z^3-3xyz$ in the conditions $0\leq x\leq y\leq z; z\geq x+1.$ We proved the following results: (i) for every positive $n,$ except for $n\equiv\pm3 \pmod9,$ $ν(n)>=1;$ (ii) for the exceptional $n,$ $ν(n)=0;$ (iii) for every prime $p\neq3,$ $ν(p)=ν(2p)=1;$ (iv) $\limsup (ν(n))=\infty;$ (v) for every positive $n,$ there exists $k$ such that $ν(k)=n.$

Positive integers: counterexample to W.M. Schmidt's conjecture

We show that there exist real numbers $α_1,α_2$ linearly independent over $\mathbb{Z}$ together with 1 such that for every non-zero integer vector $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$ one has $||m_1α_1+m_2α_2|| \ge 2^{-300} (\max(m_1, m_2))^{-σ}$ with $σ= 1.94696^+$.

On the number of pairs of positive integers $\mathbf{x, y \leq H}$ such that $\mathbf{x^2+y^2+1}$, $\mathbf{x^2+y^2+2}$ are square-free

In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also give an asymptotic formula for the number of pairs of positive integers $x, y \leq H$ such that $x^2+y^2+1$, $x^2+y^2+2$ are square-free.

On a Conjecture on the Representation of Positive Integers as the Sum of Three Terms of the Sequence $\left\lfloor \frac{n^2}{a} \right\rfloor$

We prove some cases of a conjecture by Farhi on the representation of every positive integer as the sum of three terms of the sequence $ \left\lfloor\frac{n^2}{a}\right\rfloor$. This is done by generalizing a method used by Farhi in his original paper.

On the $P_1$ property of sequences of positive integers

It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence. Certainly, there is a proof based on the Chinese Remainder Theorem. In this paper we give proofs of two analytic criteria revealing this property of sequences.

Representation of positive integers by the form $x_1...x_k + x_1 + ... + x_k$

For an arbitrary given $k\geq3,$ we consider a possibility of representation of a positive number $n$ by the form $x_1...x_k+x_1+...+x_k, 1\leq x_1\leq ... \leq x_k.$ We also study a question on the smallest value of $k\geq3$ in such a representation.

On the series of the reciprocals lcm's of sequences of positive integers: A curious interpretation

In this paper, we prove the following result: {quote} Let $\A$ be an infinite set of positive integers. For all positive integer $n$, let $τ_n$ denote the smallest element of $\A$ which does not divide $n$. Then we have $$\lim_{N \to + \infty} \frac{1}{N} \sum_{n = 1}^{N} τ_n = \sum_{n = 0}^{\infty} \frac{1}{\lcm\{a \in \A | a \leq n\}} .$${quote} In the two particular cases when $\A$ is the set of all positive integers and when $\A$ is the set of the prime numbers, we give a more precise result for the average asymptotic behavior of ${(τ_n)}_n$. Furthermore, we discuss the irrationality of th…

On certain sums concerning the gcd's and lcm's of $k$ positive integers

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordellès (2007), and extend certain related results of Hilberdink and Tóth (2016). We also formulate some conjectures and open problems.

Values at non-positive integers of generalized Euler-Zagier multiple zeta-functions

We give new closed and explicit formulas for "multiple zeta values" at non-positive integers of generalized Euler-Zagier multiple zeta-functions. We first prove these formulas for a small convenient class of these multiple zeta-functions and then use the analyticity of the values on the parameters defining the multiple zeta-functions to deduce the formulas in the general case. Also, for our aim we prove an extension of "Raabe's lemma" due to E. Friedman and A. Pereira.

On partitions of integers with restrictions involving squares

In this paper, we study partitions of positive integers with restrictions involving squares. We mainly establish the following two results (which were conjectured by Sun in 2013): (i) Each positive integer $n$ can be written as $n=x+y+z$ with $x,y,z$ positive integers such that $x^2+y^2+z^2$ is a square, unless $n$ has the form $n=2^{a}3^{b}$ or $2^{a}7$ with $a$ and $b$ nonnegative integers. (ii) Each integer $n>7$ with $n\not=11,14,17$ can be written as $n=x+y+2z$ with $x,y,z$ positive integers such that $x^2+y^2+2z^2$ is a square.

On representations of positive integers by $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \bigl(k(b+d) \bigr)^{1/3} y$, and $\bigl(k(a+c) \bigr)^{1/3} x + l(b+d) y$

We use sums of Liouville type to count the number of ways a positive integer can be represented by the forms $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \bigl(k(b+d) \bigr)^{1/3} y$, and $\bigl(k(a+c) \bigr)^{1/3} x + l(b+d) y$ for nonnegative integers $a,b,c,d,k,l,x,y$ under certain relative primality conditions.

Positive Integer Solutions of Certain Diophantine Equations Involving Lucas-Balancing numbers

In this article the Diophantine equations of the form $x^{2}-8C_{n}xy+ 16y^{2}=\pm2^{r}$, $x^{4}-8C_{n}x^{2}y+ 16y^{2}=\pm2^{r}$ and $x^{2}-8C_{n}xy^{2}+ 16y^{4}=\pm2^{r}$ are taken into consideration for the investigations of existence of any positive integer solutions. Furthermore, the solutions are manifested in terms of the balancing, Pell and Lucas-Pell numbers.

On values of the higher derivatives of the Barnes zeta function at non-positive integers

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s ζ_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbb Q(x)$, where $ζ_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-p…

关于具有至少两个不同素因子的整数模的划分函数的Newman猜想

Newman's conjecture for the partition function modulo integers with at least two distinct prime divisors

本文证明了在素因子个数为d的正整数集合中，满足划分函数Newman猜想的正整数M的密度为1，并将该类猜想推广至弱全纯模形式，应用于t-core划分与广义Frobenius划分。

A proof of the positive density conjecture for integer Apollonian circle packings

A bounded Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In this paper, we compute a lower bound for the number $κ(P,X)$ of integers less than $X$ occurring as curvatures in a bounded integer ACP $P$, and prove a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the ratio $κ(P,X)/X$ is great…

正特征下互质w-的S-整数和S-整理想的计数

Counting w-coprime S-integers and S-integral ideals in positive characteristic

本文结合解析方法、Riemann-Roch定理和Weil定理，在正特征函数域中计数互质w-的S-整数和S-整理想。

Every integer greater than 454 is the sum of at most seven positive cubes

A long-standing conjecture states that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on the basis of extensive calculations performed by the famous computationalist Zacharias Dase. We complete the proof of this conjecture, building on previous work of Linnik, Watson, McCurley, Ramaré, Boklan, Elkies, and many others.

Multiple zeta functions of Kaneko-Tsumura type and their values at positive integers

Kaneko and Tsumura introduced a new kind of multiple zeta functions $η(k_1,\ldots,k_r;s_1,\ldots,s_r)$. This is an analytic function of complex variables $s_1,\ldots,s_r$, while $k_1,\ldots,k_r$ are non-positive integer parameters. In this paper, we first extend this function to an analytic function $η(s'_1,\ldots,s'_r;s_1,\ldots,s_r)$ of $2r$ complex variables. Then we investigate its special values at positive integers. In particular, we prove some linear relations among these $η$-values and the multiple zeta values $ζ(k_1,\ldots,k_r)$ of Euler-Zagier type.

Positive Integer Solutions of the Pell Equation $x^{2}-dy^{2}=N,$ $% d\in \left\{k^{2}\pm 4,\text{}k^{2}\pm 1\right\} $ and $N\in \left\{\pm 1,\pm 4\right\}

Let $\ k$ be a natural number and $d=k^{2}\pm 4$ or $k^{2}\pm 1$. In this paper, by using continued fraction expansion of $\sqrt{d},$ we find fundamental solution of the equations $x^{2}-dy^{2}=\pm 1$ and we get all positive integer solutions of the equations $x^{2}-dy^{2}=\pm 1$ in terms of generalized Fibonacci and Lucas sequences. Moreover, we find all positive integer solutions of the equations $x^{2}-dy^{2}=\pm 4$ in terms of generalized Fibonacci and Lucas sequences. Although some of the results are well known, we think our method is elementary and different from the others.

关于包含前n个正整数不同倍数的区间长度的论文

On the length of an interval that contains distinct multiples of the first $n$ positive integers

本文证实了Erdős和Pomerance的猜想，证明存在特定长度的区间不含前n个正整数的不同倍数。

Integers represented by positive-definite quadratic forms and Petersson inner products

Let $Q$ be a positive-definite quaternary quadratic form with integer coefficients. We study the problem of giving bounds on the largest positive integer $n$ that is locally represented by $Q$ but not represented. Assuming that $n$ is relatively prime to $D(Q)$, the determinant of the Gram matrix of $Q$, we show that $n$ is represented provided that \[ n \gg \max \{ N(Q)^{3/2 + ε} D(Q)^{5/4 + ε}, N(Q)^{2 + ε} D(Q)^{1 + ε} \}. \] Here $N(Q)$ is the level of $Q$. We give three other bounds that hold under successively weaker local conditions on $n$. These results are proven by bounding the Peter…

On the Interval [n,2n]: Primes, Composites and Perfect Powers

In this paper we show that for every positive integer $n$ there exists a prime number in the interval $[n,9(n+3)/8]$. Based on this result, we prove that if $a$ is an integer greater than 1, then for every integer $n>14.4a$ there are at least four prime numbers $p$, $q$, $r$, and $s$ such that $n<ap<3n/2<aq<2n$ and $n<r<3n/2<s<2n$. Moreover, we also prove that if $m$ is a positive integer, then for every positive integer $n>14.4/(|\sqrt[m]{1.5}|-1)^m$ there exist a positive integer $a$ and a prime number $s$ such that $n<a^m<3n/2<s<2n$, as well as the fact that for every positive integer $n>14…

利用自动 Dirichlet 级数在正奇数整数处求值 $\zeta(s)$

Evaluating $ζ(s)$ At Odd Positive Integers Using Automatic Dirichlet Series

本文用 Thue-Morse 和折纸序列构造 Dirichlet 级数，求值正奇数处 zeta 函数，并给出 2015 年结果的替代证明。

A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function $σ_x(n)$

For a finite sequence of positive integers $A=\{a_j\}_{j=1}^{k},$ we prove a recursion for divisor function $σ_{x}^{(A)}(n)=\sum_{d|n,\enskip d\in A}d^x.$ As a corollary, we give an affirmative solution of the problem posed in 1969 by D. B. Lahiri [3]: to find an identity for divisor function $σ_x(n)$ similar to the classic pentagonal recursion in case of $x=1.$