Approximating reals by sums of rationals

Rational exponential sums over the divisor function

We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $τ(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $ω(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.

有理指数和的均值定理

Mean value theorems for rational exponential sums

本文利用 Mordell 的经典论证并结合其他思想，建立了 Weyl 和多种均值定理近期结果的有限域类比，而非依赖 Vinogradov 均值定理。

Sums of $L$-functions over the rational function field

Friedberg, Hoffstein and Lieman have constructed two related multiple Dirichlet series from quadratic and higher-order $L$-functions and Gauss sums. We compute these multiple Dirichlet series explicitly in the case of the rational function field. This is done by utilizing the functional equation of the $L$-functions and the functional equation relating the two multiple Dirichlet series. We also point out a very simple correspondence between these series and their $p$-parts.

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 $k$.

On error sums formed by rational approximations with split denominators

In this paper we consider error sums of the form \[\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_mα- \frac{a_m}{c_m}\,\Big) \,,\] where $α$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m ={(-1)}^m$. In particular, we investigate such sums for \[α\in \big\{ π, e,e^{1/2},e^{1/3},\dots, \log (1+t), ζ(2), ζ(3) \big\} \] and exhibit some connections between rational coefficients occurring in error sums for Apéry's continued fraction for $ζ(2)$ and well-known integer sequences. The concept of the paper generalizes the theory of ordinary error sums, which are…

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。

Sums of four rational squares with certain restrictions

In this paper we mainly study sums of four rational squares with certain restrictions. Let $\mathbb Q_{\ge0}$ be the set of nonnegative rational numbers. We establish the following four-square theorem for rational numbers: For any $a,b,c,d\in\mathbb Q_{\ge0}$, each $r\in\mathbb Q_{\ge0}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Q_{\ge0}$ such that $ax+by+cz+dw$ is a rational square (or a rational cube). This paper also contains many conjectures; for example, for any positive integers $a$ and $b$ with $\gcd(a,b)=1$, we conjecture that each $r\in\mathbb Q_{\ge0}$ can be writte…

Counting functions for sums of rational powers of integers

Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to a given $w>0$ is determined. The counting functions developed are framed in terms of convolution exponentials, and are closely related to the Riemann zeta function. At the conclusion, several estimates are derived, with special emphasis on the case of sums of square roots, i.e. $j=1$, $k=2$.

有理函数的混合不完整特征和与其平滑模数

Mixed incomplete character sums of rational functions with smooth moduli

本文研究了有理函数的混合不完整特征和，改进了平滑参数的界，并探讨了其推广与应用。

On the maximum of cotangent sums related to the Riemann Hypothesis in rational numbers in short intervals

Cotangent sums play a significant role in the Nyman-Beurling criterion for the Riemann Hypothesis. Here we investigate the maximum of the values of these cotangent sums over various sets of rational numbers in short intervals.

Some evaluation of parametric Euler sums

In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the analytic representa- tions of parametric Euler sums that involve harmonic numbers through zeta values and rational function series, either linearly or nonlinearly. Furthermore, we give explicit formulae for several parametric quadratic and cubic sums in terms of zeta values and rational series. Moreover, some interesting new consequences and illustrative examples are considered.

关于埃尔德什 - 格雷厄姆单位分数问题的拉伸指数界

Approximating $1$ by Sums of Unit Fractions

研究正整数有限多重集\(A\)，埃尔德什和格雷厄姆曾给出相关结论并提问，本文证明了对于足够大的\(K\)，\(\varepsilon(A)\leq \exp\bigl(-c\sqrt{K\log K}\bigr)\)这一拉伸指数估计。

Sums of squares of polynomials with rational coefficients

We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions with rational coefficients. In the case of ternary quartics, we prove that our counterexamples to Sturmfels' question are the only ones.

Vector-Valued Rademacher Sums and Automorphic Integrals

We present bases for certain spaces of meromorphic vector-valued rational-weight mock modular forms constructed using Rademacher sums.

Approximating rational points on toric varieties

Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if $X$ is a $\mathbb{Q}$-factorial terminal split toric variety of arbitrary dimension, then $P$ is better approximated by points on a rational curve than by any Zariski dense sequence.

Explicit and efficient formulas for the lattice point count in rational polygons using Dedekind-Rademacher sums

We give explicit, polynomial-time computable formulas for the number of integer points in any two-dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind-Rademacher sums, which are polynomial-time computable finite Fourier series. As a by-product we rederive a reciprocity law for these sums due to Gessel, which generalizes the reciprocity law for the classical Dedekind sums. In addition, our approach shows that Gessel's reciprocity law is a special case of the one for Dedekind-Rademach…

最大化有理值和或零和的数量

Maximizing the number of rational-value sums or zero-sums

本文研究n个无理数集合中r项有理值和的最大数量，确定了r<4或r≥n/2等情形的最大值，并发现该问题等价于整数序列r项零和子序列最大数量问题，是Erdős-Ginzburg-Ziv定理的变体。

逼近曲面上的有理点

Approximating rational points on surfaces

本文研究代数簇上有理点的最佳逼近问题，提出由 Vojta 猜想推出相关较弱猜想的策略，并在分裂曲面上完成完整猜想的论证。

On some properties of patial sums

The paper gives some criteria for partial sums of rational number sequences to be not rational functions and to be not algebraic functions. As an application, we study partial sums of some famous rational number sequences in mathematical analysis with relevance to number theory.

The fractional parts of Dedekind sums

We show that each rational number $r$, $0\le r<1$, occurs as the fractional part of a Dedekind sum $S(m,n)$.

Igusa's conjecture for exponential sums: optimal estimates for non-rational singularities

We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa's conjecture on exponential sums, with the log-canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for non-rational singularities, by comparing the log canonical threshold with a local notion of the motivic oscillation index.

Binary forms as sums of two squares and Châtelet surfaces

The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Châtelet surfaces defined over the rationals.

On Mordell-Tornheim sums and multiple zeta values

We prove that any Mordell-Tornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, it follows that any Mordell-Tornheim sum with weight and depth of opposite parity can be expressed as a rational linear combination of products of multiple zeta values of lower depth.

Approximating elements of the middle third Cantor set with dyadic rationals

Let $C$ be the middle third Cantor set and $μ$ be the $\frac{\log 2}{\log 3}$-dimensional Hausdorff measure restricted to $C$. In this paper we study approximations of elements of $C$ by dyadic rationals. Our main result implies that for $μ$ almost every $x\in C$ we have $$\#\left\{1\leq n\leq N:\left|x-\frac{p}{2^n}\right| \leq \frac{1}{n^{0.01}\cdot 2^{n}}\textrm{ for some }p\in\mathbb{N}\right\}\sim 2\sum_{n=1}^{N}n^{-0.01}.$$ This improves upon a recent result of Allen, Chow, and Yu which gives a sub-logarithmic improvement over the trivial approximation rate.

Explicit evaluation of certain sums of multiple zeta-star values

Bowman and Bradley proved an explicit formula for the sum of multiple zeta values whose indices are the sequence (3,1,3,1,...,3,1) with a number of 2's inserted. Kondo, Saito and Tanaka considered the similar sum of multiple zeta-star values and showed that this value is a rational multiple of a power of π. In this paper, we give an explicit formula for the rational part. In addition, we interpret the result as an identity in the harmonic algebra.

Bounding the rational sums of squares over totally real fields

Let K be a totally real Galois number field. C. J. Hillar proved that if f in Q[x_1,...,x_n] is a sum of m squares in K[x_1,...,x_n], then f is a sum of N(m) squares in Q[x_1,...,x_n]. Modifying Hillar's proof, we improve the improve the bound given for N(m), the proof being constructive as well.

有理权全纯爱森斯坦级数与伽马函数特殊值

Holomorphic Eisenstein series of rational weights and special values of Gamma function

本文构造了 $Γ_0(p)$ 上权大于 2 的有理权全纯爱森斯坦级数，证明其为模形式并给出傅里叶展开，进而建立与有理权 eta 商的四类恒等式，最终导出伽马函数在有理点处特殊值的级数表达式。

有理立方和与3-塞尔默（Selmer）群

Sums of rational cubes and the $3$-Selmer group

本研究得到j不变量为0的椭圆曲线三次扭族中3-塞尔默群的分布，改进了两有理立方和整数密度的上下界，通过强化解析核心、证明三线性大筛法解决了此前方法的挑战。

逼近球对称簇上的有理点

Approximating rational points on horospherical varieties

该研究在Vojta猜想成立的前提下，构造过球对称簇上有理点的曲线以实现更优的有理点逼近，证明了McKinnon猜想在球对称情形的弱化版本，结果可推广至带末端奇点的Q-阶乘球对称簇。

有理数中的单色积与和

Monochromatic products and sums in the rationals

证明了在有理数的任意有限多色着色中，必存在某种颜色包含由非零 $x$ 和 $y$ 构成的形如 $\{x,y,xy,x+y\}$ 的集合。

空隙序列及其倒数和表示区间内所有有理数

Lacunary sequences whose reciprocal sums represent all rational numbers in an interval

论文推翻Bleicher和Erdős的猜想，证明存在空隙序列，其有限和可表示区间内所有有理数，并研究了更强的变体问题。

Sums of finitely many distinct rationals

${\cal E}$ denotes the family of all finite nonempty $S\subseteq{\mathbb N}:=\{1,2,\ldots\}$, and ${\cal E}(X):={\cal E}\cap\{S:S\subseteq X\}$ when $X\subseteq{\mathbb N}$. Similarly, ${\cal F}$ denotes the family of all finite nonempty $T\subseteq{\mathbb Q}^+$, and ${\cal F}(Y) := {\cal F}\cap\{T:T\subseteq Y\}$ where ${\mathbb Q}^+$ is the set of all positive rationals and $Y\subseteq{\mathbb Q}^+$. This paper treats the functions $σ:{\cal E}\rightarrow{\mathbb Q}^+$ given by $σ:S\mapstoσS :=\sum\{1/x:x\in S\}$, the function $δ:{\cal E}\rightarrow{\mathbb N}$ defined by $σS = νS/δS$ where …

Sums of Digits in q-ary Expansions

Let $s_q(n)$ denote the sum of the digits of a number $n$ expressed in base $q$. We study here the ratio $\frac{s_q(n^α)}{s_q(n)}$ for various values of $q$ and $α$. In 1978, Kenneth B. Stolarsky showed that $\lim\inf_{n\rightarrow\infty}\frac{s_2(n^2)}{s_2(n)}=0$ and that $\lim\sup_{n\rightarrow\infty}\frac{s_2(n^2)}{s_2(n)}=\infty$ using an explicit construction. We show that for $α=2$ and $q\geq 2$, the above ratio can in fact be any positive rational number. We also study what happens when $α$ is a rational number that is not an integer, terminating the resulting expression by using the fl…

Binomial sums related to rational approximations to $ζ(4)$

For the solution $\{u_n\}_{n=0}^\infty$ to the polynomial recursion $(n+1)^5u_{n+1}-3(2n+1)(3n^2+3n+1)(15n^2+15n+4)u_n -3n^3(3n-1)(3n+1)u_{n-1}=0$, where $n=1,2,...$, with the initial data $u_0=1$, $u_1=12$, we prove that all $u_n$ are integers. The numbers $u_n$, $n=0,1,2,...$, are denominators of rational approximations to $ζ(4)$ (see math.NT/0201024). We use Andrews's generalization of Whipple's transformation of a terminating ${}_7F_6(1)$-series and the method from math.NT/0311114.

具有数字受限分母的有理逼近

Rational approximation with digit-restricted denominators

本文证明了实数γ的“良好”逼近存在性，使用基b下由数字0和1构成分母的有理数，推导了基本估计并通过处理指数和增强该结果。

二元三次型与有理数立方和问题

Binary Cubic Forms and Rational Cube Sum Problem

本文利用整二元三次型研究有理数立方和问题，无条件证明了在特定剩余类中存在无穷多个素数可表示为两个有理数的立方和。

Sums of four and more unit fractions and approximate parametrizations

We prove new upper bounds on the number of representations of rational numbers $\frac{m}{n}$ as a sum of $4$ unit fractions, giving five different regions, depending on the size of $m$ in terms of $n$. In particular, we improve the most relevant cases, when $m$ is small, and when $m$ is close to $n$. The improvements stem from not only studying complete parametrizations of the set of solutions, but simplifying this set appropriately. Certain subsets of all parameters define the set of all solutions, up to applications of divisor functions, which has little impact on the upper bound of the numb…

Generic A-family of exponential sums

In this paper we construct a generating polynomial over the rationals for the generic Newton polygon for the L function of exponential sums of the family of f = x^d+ a x^s parameterized by a, and prove some of its key properties. The generating polynomial encodes information of and determines the generic Newton polygon at each prime p when p is large enough, and vice versa.

Representing multiples of $m$ in real quadratic fields as sums of squares

We study real quadratic fields $\mathbb{Q}(\sqrt{D})$ such that, for a given rational integer $m$, all $m$-multiples of totally positive integers are sums of squares. We prove quite sharp necessary and sufficient conditions for this to happen. Further, we give a fast algorithm that solves this question for specific $m$, $D$ and we give complete results for $m \leq 5000$.

The Frobenius problem, rational polytopes, and Fourier-Dedekind Sums

We study the number of lattice points in integer dilates of the rational polytope $P = (x_1,...,x_n) \in \R_{\geq 0}^n : \sum_{k=1}^n x_k a_k \leq 1$, where $a_1,...,a_n$ are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers $a_1,...,a_n$, find the largest value of t (the Frobenius number) such that $m_1 a_1 + ... + m_n a_n = t$ has no solution in positive integers $m_1,...,m_n$. This is equivalent to the problem of finding the largest dilate tP such that the facet $\sum_{k=1}^n x_k a_k = t$ contains no …

Long nonnegative sums of Legendre symbols

For $0\leq α<1$ and prime number $p$ let $L(α,p)$ be the sum of the first $[αp]$ values of Legendre symbol modulo $p$. We study positivity of $L(α,p)$ and prove that for $|α-\frac13|<2\cdot 10^{-6}$ and for rational $α\leq \frac12$ with denominators in the set $\{1,2,3,4,5,6,8,12\}$ the inequality $L(α,p)\geq 0$ holds for majority of primes.

On the distribution of Ramanujan Sums over number fields

For a number field $\mathbb{K}$, and integral ideals $\mathcal{I}$ and $\mathcal{J}$ in its number ring $\mathcal{O}_{\mathbb{K}}$, Nowak studied the asymptotic behaviour of the average of Ramanujan sums $C_{\mathcal{J}}({\mathcal{I}})$ over both ideals $\mathcal{I}$ and $\mathcal{J}$. In this article, we extend this investigation by establishing asymptotic formulas for the second moment of averages of Ramanujan sums over quadratic and cubic number fields, thereby generalizing previous works of Chen, Kumchev, Robles, and Roy on moments of averages of Ramanujan sums over rationals. Additionally…

Stratification for multiplicative character sums

We prove a stratification result for certain families of $n$-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of $r$ multiplicative characters evaluated at rational functions, and the families (with $nr$ parameters) are obtained by allowing each of the $r$ rational functions to be replaced by an "offset", i.e. a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least $j\lfloor(r-1)/2(n-1)\rfloor$ for $1\le …

Some sums over irreducible polynomials

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form $\sum_P h(P)$ where the sum is over monic irreducible polynomials $P$ in $\mathbb{F}_q[T]$, the function $h$ is a rational function and the sum is considered in the $T^{-1}$-adic topology. As an example of our results, in $\mathbb{F}_2[T]$, the sum $\sum_P \tfrac{1}{P^k - 1}$ always converges to a rational function, and is $0$ for $k=1$.

关于Dedekind和的受限平均

On restricted averages of Dedekind sums

本文研究固定$α\leq 1/2$时特定有理数集合上Dedekind和的平均值，将先前$α=1/2$的渐近结果扩展至所有固定有理数及几乎所有无理数$α$，并量化了by-excess Euclidean算法运行时间渐近式第二项的偏差。

A conjecture in the problem of rational definite summation

A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.

关于Nahm和的Zagier对偶猜想的反例

Counterexamples to Zagier's Duality Conjecture on Nahm Sums

针对Nahm问题中Zagier提出的模形式Nahm和的对偶也为模形式的猜想，构造出秩为4的模形式Nahm和反例，其对偶不具备模形式性质，从而证伪该对偶猜想。

Gosper Summability of Rational Multiples of Hypergeometric Terms

By telescoping method, Sun gave some hypergeometric series whose sums are related to $π$ recently. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term $t_k$, we consider the Gosper summability of $r(k)t_k$ for $r(k)$ being a rational function of $k$. We give an upper bound and a lower bound on the degree of the numerator of $r(k)$ such that $r(k)t_k$ is Gosper summable. We also show that the denominator of the $r(k)$ can read from the Gosper representation of $t_{k+1}/t_k$. Based on these results, we give a systematic method to construct series…

Simultaneous Diophantine approximation: sums of squares and homogeneous polynomials

Let $f$ be a homogeneous polynomial with rational coefficients in $d$ variables. We prove several results concerning uniform simultaneous approximation to points on the graph of $f$, as well as on the hypersurface $\{f(x_1,\dots,x_d) = 1\}$. The results are first stated for the case $f(x_1,\dots,x_d) = x_1^2+\dots+x_d^2,$ which is of particular interest.

Supercongruences for sums involving $\binom ak^m$

Let $p$ be an odd prime, and let $a$ be a rational $p$-adic integer with $a\not\equiv 0\pmod p$. In this paper, using WZ method we establish the congruences for $\sum_{k=0}^{p-1} \binom ak^2(-1)^k(1-\frac 2ak)$ modulo $p^2$ and $\sum_{k=0}^{p-1} \binom ak^r(1-\frac 2ak)^s$ modulo $p^4$, where $r\in\{3,4\}$ and $s\in\{1,3\}$.

Improved Tail Estimates for the Distribution of Quadratic Weyl Sums

We consider quadratic Weyl sums $S_N(x;c,α)=\sum_{n=1}^N\exp\{2πi((\frac{1}{2}n^2+cn)x+αn)\}$ for $c=α=0$ (the rational case) or $(c,α)\notin\mathbb{Q}^2$ (the irrational case), where $x$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. The limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;c,α)$ as $N\to\infty$ was described by Marklof [13] (respectively Cellarosi and Marklof [5]) in the rational (resp. irrational) case. According to the limiting distribution, the probability of landing outside a ball of…

Alternating Euler sums and special values of Witten multiple zeta function attached to so(5)

In this note we shall study the Witten multiple zeta function associated to the Lie algebra so(5) defined by Matsumoto. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w>2 is a finite rational linear combination of alternating Euler sums of weight w and depth at most two, except when the only nonzero argument is one of the two last variables in which case $ζ(w-1)$ is needed.

带有理参数的光滑Theta和的界

Bounds for Smooth Theta Sums with Rational Parameters

该研究构造了一族显式实参数对，证明对足够正则的函数，带二次相位与线性相位的高维光滑theta和满足一致的$N^{k/2}$阶上界，证明借助自守函数在horocycle提升曲线上的有界性完成。

Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters

We consider quadratic Weyl sums $S_N(x;α,β)=\sum_{n=1}^N \exp\!\left[2πi\left( \left(\tfrac{1}{2}n^2+βn\right)\!x+αn\right)\right]$ for $(α,β)\in\mathbb{Q}^2$, where $x\in\mathbb{R}$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. We prove that the limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;α,β)$ as $N\to\infty$ is either heavy tailed or compactly supported, depending solely on $α,β$. In the heavy tailed case, the probability (according to the limiting distribution) of landing outside a ball of r…