Estimates of Newman Sum over Numbers Multiple of a Fixed Integer

Finite searches, Chowla's cosine problem, and large Newman polynomials

A length $n$ cosine sum is an expression of the form $\cos a_1θ+ \cdots + \cos a_nθ$ where $a_1 < \cdots < a_n$ are positive integers, and a length $n$ Newman polynomial is an expression of the form $z^{a_1} + \cdots + z^{a_n}$ where $a_1 < \cdots < a_n$ are nonnegative integers. We define $-λ(n)$ to be the largest minimum of a length $n$ cosine sum as $\{a_1,\ldots,a_n\}$ ranges over all sets of $n$ positive integers, and we define $μ(n)$ to be the largest minimum modulus on the unit circle of a length $n$ Newman polynomial as $\{a_1,\ldots,a_n\}$ ranges over all sets of $n$ nonnegative integ…

On Mex-related partition functions of Andrews and Newman

The minimal excludant, or "mex" function, on a set $S$ of positive integers is the least positive integer not in $S$. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely $p_{t,t}(n)$, and provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$. In this article, we study the parity of $p_{t,t}(n)$ when $t=2^α, 3\cdot 2^α$ for …

On Monotonic Strengthening of Newman-like Phenomenon on (2m+1)-multiples in Base 2m

We obtain exact and asymptotic expressions for the excess of nonnegative (2m+1)-multiples less than (2m)^k with even digit sums in the base 2m.

On Newman and Littlewood multiples of Borwein polynomials

A Newman polynomial has all the coefficients in $\{ 0,1\}$ and constant term 1, whereas a Littlewood polynomial has all coefficients in $\{-1,1\}$. We call $P(X)\in\mathbb{Z}[X]$ a Borwein polynomial if all its coefficients belong to $\{ -1,0,1\}$ and $P(0)\neq 0$. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle $|z|=1$ has a non-zero multiple in $\mathbb{Z}[X]$ with coefficients in a finite set $\mathcal{D} \subset \mathbb{Z}$, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or N…

On a class of lacunary almost Newman polynomials modulo p and density theorems

The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $ζ_β(z)$ of the $β$-shift, for $β> 1$ close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in $\mathbb{F}_p$ and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density th…

对Andrews和Newman关于最小排除项之和结果的一个精细化

A refinement of a result of Andrews and Newman on the sum of minimal excludants

本文精细化Andrews和Newman关于最小排除项之和的恒等式，得到相关函数的模4和模8同余式，并提出新的同余猜想，同时开始研究最小排除项的高阶矩。

Distribution of the partition function modulo m

Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the partition function hits every residue class modulo a given integer M infinitely often).

拓展安德鲁斯和纽曼对 crank-mex 定理的细化

Extending Andrews and Newman's refinement of the crank-mex theorem

本文基于 crank-mex 定理，研究偶数 mex 分拆与负 crank 和正 crank 分拆的关系，通过分析和组合方法证明了相关结果。

On Newman's phenomenon in higher bases

A well known result of Newman says that upto a limit, multiples of $3$ with even number of 1's in binary representation always exceed multiples of $3$ with odd number of 1's. The phenomenon of preponderance of even number of 1's is now known as Newman's phenomenon. We show that this phenomenon exists for higher bases. Let $b$ be a positive integer($\geq 2$). Let $A_{b}$ be the set of all natural numbers which contain only 0's and 1's in b-ary expansion and $S^{(b)}_{q,i}(n)$ be the difference between the corresponding number of $k_e<n$, $k_e\equiv i \mod q$, $k_e\in A_{b}$ and $k_e$ has even n…

Tangent power sums and their applications

For integer $m, p,$ we study tangent power sum $\sum^m_{k=1}\tan^{2p}\frac{πk}{2m+1}.$ We prove that, for every $m, p,$ it is integer, and, for a fixed p, it is a polynomial in $m$ of degree $2p.$ We give recurrent, asymptotical and explicit formulas for these polynomials and indicate their connections with Newman's digit sums in base $2m.$

Effective approximation of heat flow evolution of the Riemann $ξ$ function, and a new upper bound for the de Bruijn-Newman constant

For each $t \in \mathbf{R}$, define the entire function $$ H_t(z) := \int_0^\infty e^{tu^2} Φ(u) \cos(zu)\ du$$ where $Φ$ is the super-exponentially decaying function $$ Φ(u) := \sum_{n=1}^\infty (2π^2 n^4 e^{9u} - 3πn^2 e^{5u} ) \exp(-πn^2 e^{4u} ).$$ This is essentially the heat flow evolution of the Riemann $ξ$ function. From the work of de Bruijn and Newman, there exists a finite constant $Λ$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are all real precisely when $t \geq Λ$. The Riemann hypothesis is equivalent to the assertion $Λ\leq 0$; recently, Rodgers and Tao …

Integers representable as the product of the sum of four integers with the sum of their reciprocals

By the theory of elliptic curves, we study the integers representable as the product of the sum of four integers with the sum of their reciprocals and give a sufficient condition for the integers with a positive representation.

A sum of three nonunit squares of integers

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann Hypothesis(GRH) holds. As applications, we find all integers, under the GRH only when $k=3$, which are sums of $k$ nonzero triangular numbers, sums of $k$ nonzero generalized pentagonal numbers, and sums of $k$ nonzero generalized octagonal numbers, respectively for any integer $k\ge 3$.

三个整数，其和、积及两两乘积之和均为完全平方数

Three integers whose sum, product and the sum of the products of the integers, taken two at a time, are perfect squares

针对欧拉提出的三整数和、积及两两乘积之和均为平方数的问题，本文用新方法得到低次多项式参数化解及数值较小的解。

将整数表示为多项式立方和

Expressing an integer as a sum of cubes of polynomials

本文证明无穷多整数可表示为四个整系数多项式立方和，且每个整数可表示为五个多项式立方和，并给出1和2的具体恒等式。

On Integer Sequences Associated To Two Distinct Sums

In this note, we show the existence of integer sequences of lengths at least 3 (except 7) such that for every integer in position $i\equiv 1\pmod{4}$ (respectively position $j\equiv 3\pmod{4}$), counting from left to right, the sum of the integer and the adjacent integer(s) has a constant sum $x$ (respectively $y$) with $x\ne y$.

Smooth arithmetical sums over k-free integers

We use partial zeta functions to analyse the asymptotic behaviour of certain smooth arithmetical sums over smooth k-free integers.

On sums of S-integers of bounded norm

We prove an asymptotic formula for the number of S-integers in a number field K that can be represented by a sum of n S-integers of bounded norm.

Bernoulli numbers and sums of powers of integers of higher order

We give an expression of polynomials for higher sums of powers of integers via the higher order Bernoulli numbers.

The exponential sum over squarefree integers

We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Brüdern and Perelli.

Multiple harmonic sums and multiple harmonic star sums are (nearly) never integers

It is well known that the harmonic sum $H_n(1)=\sum_{k=1}^n\frac{1}{k}$ is never an integer for $n>1$. In 1946, Erdős and Niven proved that the nested multiple harmonic sum $H_n(\{1\}^r)=\sum_{1\le k_1<\dots<k_r\le n}\frac{1}{k_1\cdots k_r}$ can take integer values only for a finite number of positive integers $n$. In 2012, Chen and Tang refined this result by showing that $H_n(\{1\}^r)$ is an integer only for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In this paper, we consider the integrality problem for arbitrary multiple harmonic and multiple harmonic star sums and show that none of these sums is an…

关于无平方因子整数与光滑整数的 Kloosterman 和之和

Sums of Kloosterman sums over square-free and smooth integers

本文受近期关于 Kloosterman 和的双线性型及扭曲和的研究启发，研究了由无平方因子整数和光滑整数参数化的 Kloosterman 和之和的若干新问题。

An analogue of Ramanujan's sum with respect to regular integers (mod $r$)

An integer $a$ is said to be regular (mod $r$) if there exists an integer $x$ such that $a^2x\equiv a\pmod{r}$. In this paper we introduce an analogue of Ramanujan's sum with respect to regular integers (mod $r$) and show that this analogue possesses properties similar to those of the usual Ramanujan's sum.

Representation of an integer as the sum of a prime in arithmetic progression and a square-free integer

Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramaré and may be viewed as an abstract circle method.

On the sum of integers from some multiplicative sets and some powers of integers

We show that if there exists an integer subject to some congruence conditions that cannot be written as the sum of the norm of an ideal in $\mathbb{Z}[\exp(2πi/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then there are infinitely many such integers. Also, if there exists an integer that cannot be written as the sum of an integer which is the norm of an ideal in in $\mathbb{Z}[\exp(2πi/p)]$ and at most $p-2$ powers of $p$, where $p\geq 3$ is a prime, then there are infinitely many such integers. Finally it is shown that there are infinitely many integers not the sum of the norm of an ideal…

On the alternating sums of powers of consecutive q-integers

In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.

Sums of integers and sums of their squares

Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares of $m$ integers? Denoting this set of all possible $T$ by $\mathscr{S}_m(n)$, Goldmakher and Pollack have given a simple characterization of $\mathscr{S}_4(n)$ using elementary arguments. Their result can be reinterpreted in terms of Mordell's theory of representations of binary integral quadratic forms as sums of squares of integral linear f…

整数作为两个素数幂之和的平均表示数

The average number of representations of an integer as a sum of two prime powers over multiples of a fixed integer

本文扩展Ikeda和Suriajaya(2025)的结果，研究固定整数q的倍数上整数n作为两个素数k次幂之和的表示数的渐进行为。

Representations of integers as sums of primes from a Beatty sequence

We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{α,β}$, where $α>1$ is irrational and of finite type.

除固定整数外的整数平方和

Sums of squares of integers except for a fixed one

本文研究除固定整数外的整数平方和问题，对任意非负整数$n$找到了能表示所有相关正整数的最小平方数数量，推广了Dubouis在$n=0$时的结果。

The density of integers representable as the sum of four prime cubes

The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$. This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by Liu.

A note on the alternating sums of powers of consecutive q-integers

In this paper we construct a new q-Euler numbers and polynomials. By using these numbers and polynomials, we give the interesting formulae related to alternating sums of powers of consecutive q-integers following an idea due to Euler.

On integers which are representable as sums of large squares

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set $\{n^2,(n+1)^2,\ldots \}$ is asymptotically $O(n^2)$, verifying thus a conjecture of Dutch and Rickett. Furthermore we ask a question on the representation of integers as sum of four large squares.

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 $\sqrt{n}$.

Sums of squares in $S$-integers

In totally real number fields, we characterize the rings of $S$-integers (obtained by inverting a rational integer $m$) such that all their totally positive elements are represented as a sum of squares. We further obtain partial answers to the question: when are all the totally positive algebraic integers that are divisible by $m$ represented as a sum of squares?

On the mean average of integer partition as the sum of powers

This paper is concerned with the function $r_{k,s}(n)$, the number of (ordered) representations of $n$ as the sum of $s$ positive $k$-th powers, where integers $k,s\ge 2$. We examine the mean average of the function, or equivalently, \begin{equation*} \sum_{m=1}^n r_{k,s}(m). \end{equation*}

整数表示为三个或更多素数之和的表示数平均值，在固定整数倍数上的推广

The average number of representations of an integer as a sum of three or more primes, over multiples of a fixed integer

该论文将Ikeda和Suriajaya的结果推广至任意$j\ge 3$个素数之和的表示数平均值，证明一般情形下渐近公式成立，并引入广义哥德巴赫问题奇异级数的平均值作为新工具。

Parity distribution and divisibility of Mex-related partition functions

Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(λ)$ for an integer partition $λ$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $λ$. They then defined $p_{A,a}(n)$ to be the number of partitions $λ$ of $n$ satisfying $\text{mex}_{A,a}(λ)\equiv a\pmod{2A}$. They found the generating function for $p_{t,t}(n)$ and $p_{2t,t}(n)$ for any positive integer $t$, and studied their arithmetic properties for some small values of $t$. In this article, we study the partition function $p_{mt,t}(n)$ for all positive integers $m$ a…

Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

Considering the problem of finding all the integer solutions of the sum of $M$ consecutive integer squares starting at $a^{2}$ being equal to a squared integer $s^{2}$, it is shown that this problem has no solutions if $M\equiv3,5,6,7,8$ or $10 (mod\,12)$ and has integer solutions if $M\equiv0,9,24$ or $33 (mod\,72)$; or $M\equiv1,2$ or $16 (mod\,24)$; or $M\equiv11 (mod\,12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv1 (mod\,24)$ and $(M-1)/24$ are all pentagonal numbers, except the first two.

Mean representation number of integers as the sum of primes

Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best possible.

关于整数表示为无平方因子数与特殊类型素数之和

On the representation of integer as sum of a square-free number and a prime of special type

本文证明存在无穷多个整数可表示为一个无平方因子整数与一个满足特定丢番图不等式条件的素数之和，其中系数α为无理数。

Integers that are sums of two rational sixth powers

We prove that $164634913$ is the smallest positive integer that is a sum of two rational sixth powers but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$, we use the existence of morphisms from $C_{k}$ to elliptic curves, together with the Mordell-Weil sieve, to rule out the existence of rational points on $C_{k}$ for various $k$.

Decomposition of an integer as the sum of two cubes to a fixed modulus

The representation of any integer as the sum of two cubes to a fixed modulus is always possible if and only if the modulus is not divisible by seven or nine. For a positive non-prime integer N there is given an inductive way to find its remainders that can be represented as the sum of two cubes to a fixed modulus N. Moreover, it is possible to find the components of this representation.

Signed harmonic sums of integers with $k$ distinct prime factors

We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.

On a conjecture for representations of integers as sums of squares and double shuffle relations

In this paper, we prove a conjecture of Chan and Chua for the number of representations of integers as sums of 8s integral squares. The proof uses a theorem of Imamoglu and Kohnen, and the double shuffle relations satisfied by the double Eisenstein series of level 2.

Representing Integers as the Sum of Two Squares in the Ring $\Z_n$

A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime factorization of $z$. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the representation of $z$ as the sum of two squares. Viewing each of these questions in $\Z_n$, the ring of integers modulo $n$, we give a characterization of all integers $n\ge 2$ such that every $z\in \Z_n$ can be written as the sum of two squares…

Lucas-Carmichael 整数与基-$p$ 位数和的一个新刻画

A new characterization for the Lucas-Carmichael Integers and sums of base-$p$ digits

本文通过基-$p$ 位数和给出 Lucas-Carmichael 整数的充分必要刻画，研究其相关性质，并在素数 $k$-元组猜想下证明其无穷性。

Difference in the Number of Summands in the Zeckendorf Partitions of Consecutive Integers

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. We study the difference between the number of summands in the partition of two consecutive integers. In particular, let $L(n)$ be the number of summands in the partition of $n$. We characterize all positive integers such that $L(n) > L(n+1)$, $L(n) < L(n+1)$, and $L(n) = L(n+1)$. Furthermore, we call $n+1$ a peak of $L$ if $L(n) < L(n+1) > L(n+2)$ and a divot of $L$ if $L(n) > L(n+1) < L(n+2)$. We characterize all such peaks and divots of $L$.

∑_{i=1}^{n}1/gcd(n,i) 的整数值

The Integer Values of $\sum_{i=1}^{n}\frac{1}{\gcd(n,i)}$

该研究探讨使∑_{i=1}^{n}1/gcd(n,i)取整数值的n≤x的集合，证明这类整数的数量为≪x^{2/3+ε}。

Cube sum problem for integers having exactly two distinct prime factors

Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to the works of Sylvester, Satgé, Selmer etc. and up to the recent works of Alpöge-Bhargava-Shnidman. In this article, we consider the cube sum problem for cube-free integers n which has two distinct prime factors none of which is 3.

有理整数作为单位之和——二次情况

Rational integers as sums of units -- the quadratic case

本文研究了二次数域中自然数作为k个单位和的渐进行为，解决了Jarden和Narkiewicz在2007年提出的问题。

On sets of integers which are both sum-free and product-free

We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.

实双二次域上整元素整数倍数的平方和

Sums of squares of integer-multiple of an integral element on real bi-quadratic fields

本文在实双二次域中构造全正代数整数，给出其正整数倍不能表示为整数平方和的必要条件，并分两类整数以表格示例验证。

On the Structure of Additive Systems of Integers

A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an analogous concept where only sums of elements are considered. We establish a bijection between sum systems and sum-and-distance systems of corresponding size, and show that sum systems are equivalent to principal reversible cuboids, which are tensors…