Note on large quadratic character sums
Zikang Dong, Yutong Song, Ruihua Wang, Shengbo Zhao
本文在广义黎曼假设下，证明了二次狄利克雷特征和存在大值的Omega结果，深化了对该特征和极值行为的理解。
本文研究二次狄利克雷特征和的条件大值。在广义黎曼假设下，我们证明了二次特征和的一个Omega结果。
In this article, we investigate the conditional large values of quadratic Dirichlet character sums. We prove an Omega result for quadratic character sums under the assumption of the generalized Riemann hypothesis.
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In this paper we consider Dedekind type DC sums and prove receprocity laws related to DC sums.
Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.
it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.
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.
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients.
In this paper we construct the q-analogue of Barnes' Bernoulli numbers and plynomials of degree 2, which is an answer to a part of Schlosser's question. Finally, we treat the q-analogue of the sums of powers of consecutive integrs.
Kannan Soundararajan recently obtained a new estimate, conditional to the Riemann hypothesis, for the summatory function of the Mobius function. In this expository article we describe his method, with detailed computations.
In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} ϕ\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{ϕ([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. and Wu.
The Ohno relation is a well-known relation among multiple zeta values. Hirose, Onozuka, Sato, and the author investigated the sum related to the Ohno relation and presented two types of new relations and five conjectural formulas. This paper proves one of these formulas.
A Note on Sun's Conjectures on Apéry-like Sums Involving Lucas Sequences and Harmonic Numbers
本文利用Davydychev--Kalmykov的相关结果，证明了Zhi-Wei Sun提出的关于涉及Lucas序列与调和数的类Apéry和的四个猜想恒等式。
Following the work of Hildebrand we improve the Po'lya- Vinogradov inequality in a specific range, we also give a general result that shows its dependency on Burgess bound and at last we improve the range of validity for a special case of Burgess' character sum estimate.
In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is still open. Finally, we will treat the $q$-analogue of the sums of powers of consecutive integers.
Recently, the explicit evaluation of Gauss sums in the index 2 and 4 cases have been given in several papers (see [2,3,7,8]). In the course of evaluation, the sigh (or unit root) ambiguities are unavoidably occurred. This paper presents another method, different from [7] and [8], to determine the sigh (unit root) ambiguities of Gauss sums in the index 2 case, as well as the ones with odd order in the non-cyclic index 4 case. And we note that the method in this paper are more succinct and effective than [8] and [7].
In the present paper, our objective is to treat a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight in connection with modified q-Genocchi polynomials with weight alpha by using p-adic invariant q-integral on Zp.
In 1991, Baker and Harman proved, under the assumption of the generalized Riemann hypothesis, that $\max_{ θ\in [0,1) }\left|\sum_{ n \leq x } μ(n) e(n θ) \right| \ll_εx^{3/4 + ε}$. The purpose of this note is to deduce an analogous bound in the context of polynomials over a finite field using Weil's Riemann Hypothesis for curves over a finite field. Our approach is based on the work of Hayes who studied exponential sums over irreducible polynomials.
A note on the coefficients of power sums of arithmetic progressions
本文提出了一个仅含二项式系数和二项式幂的双重求和公式，用于计算任意等差数列幂和相关多项式的系数，并给出了涉及第一类和第二类 Stirling 数的 Bernoulli 多项式的显式公式。
For a given positive integers $m$ and $\ell$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2 \leq n \leq m$, there are distinct $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$.
We prove an upper bound for the exponential sum associated to a localized $k-$divisor function, i.e., the counting function of the number of ways to write a positive integer $n$ as a product of $k\ge 2$ positive integers, each of them belonging to a specified interval. In particular, this gives an estimate for the exponential sum for the $k-$divisor function, $d_k(n)$.
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta values, and the Witten zeta function. We discuss a heuristic for finding or dismissing the existence of similar simple sums. We also produce some new sums from recursions involving the Riemann zeta and the Dirichlet beta functions.
A note on The asymptotic uniform distribution of subset sums
本文通过Li和Wan提出的显式公式，简化证明了子集和的渐近均匀分布定理。
This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.
This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.
Note on Fractional Sums with Fixed GCD
本文研究了具有固定最大公约数的分数和的渐近公式，分析了算术函数在不同整数乘积下的表示个数及其相关性质。
In this note, we deduce an asymptotic formula for even power moments of Kloosterman sums based on the important work of N. M. Katz on Kloosterman sheaves. In a similar manner, we can also obtain an upper bound for odd power moments. Moreover, we shall give an asymptotic formula for odd power moments of absolute Kloosterman sums. Consequently, we find that there are infinitely many $a\bmod p$ such that $S(a,1;p)\gtrless0$ as $p\rightarrow+\infty.$
In this note, we first review the novel approach to power sums put forward recently by Muschielok in arXiv:2207.01935v1, which can be summarized by the formula $S_m^{(a)}(n) = \sum_{k} c_{mk} ψ_k^{(a)}(n)$, where the $c_{mk}$'s are the expansion coefficients and where the basis functions $ψ_m^{(a)}(n)$ fulfil the recursive property $ψ_m^{(a+1)}(n)= \sum_{i=1}^n ψ_m^{(a)}(i)$. Then, we point out a number of supplementary facts concerning the said approach not contemplated explicitly in Muschielok's paper. In particular, we show that, for any given $m$, the values of the $c_{mk}$'s can be obtain…
A note on log-type GCD sums and derivatives of the Riemann zeta function
本文证明了对数型GCD和的上界，推广了Gál定理，并分别用无条件的谱范数方法和基于黎曼zeta函数导数条件上界的方法给出两种证明。
We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $ζ(s,α)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $ζ(s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to0$.
We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.
A note on the distribution of clusters and deserts of prime numbers
本文利用三角多项式值分布公式，研究加权 von Mangoldt 函数值之和的分布，并给出其服从类正态分布的证据。
In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.
Solutions to some sign change problems on the functions involving sums of divisors
本文针对 Pongsriiam 近期提出的涉及除数和函数的符号变化问题给出解答，填补了相关数论问题的研究空白。
A Note on Large Sums of Divisor-Bounded Multiplicative Functions
本文研究除数有界乘性函数部分和的下界传递性质，证明两个满足特定下界条件的乘性函数之积也能在更大范围内获得相应的部分和下界。
Let $A\subset\mathbb{N}$, $α\in(0,1)$, and for $x\in\mathbb{R}$ let $e(x):=e^{2πix}$. We set $$S_{A}(α,N):=\sum_{\substack{n\in A\n\leq N}}e(nα).$$ Recently, Lambert A'Campo proposed the following question: is there an infinite non-cofinite set $A\subset\mathbb{N}$ such that for all $α\in(0,1)$ the sum $S_{A}(α,N)$ has bounded modulus as $N\to +\infty$? In this note we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum $S_{A}(α,N)$ is bounded in modulus on an arbitrarily small interval an…
In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl's bound for exponential sums and a device used by Popov.
A prime number $p$ is called a Schenker prime if there exists such $n\in\mathbb{N}_+$ that $p\nmid n$ and $p\mid a_n$, where $a_n = \sum_{j=0}^{n}\frac{n!}{j!}n^j$ is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning $p$-adic valuations of $a_n$ in case when $p$ is a Schenker prime. In particular, they asked whether for each $k\in\mathbb{N}_+$ there exists the unique positive integer $n_k<p^k$ such that $v_p(a_{m\cdot 5^k + n_k})\geq k$ for each nonnegative integer $m$. We prove that for every $k\in\mathbb{N}_+$ the inequality $v_5(a_n)\geq k$ h…
We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.
In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.
Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1} S_{k+i}(n), \end{equation*} where $S_k^{(0)}(n) \equiv S_k(n)$ is the ordinary power sum $1^k + 2^k + \cdots + n^k$. In this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly a…
Problems 66 and 67 on sums of residue classes and primes of András Sárközy's collection of unsolved problems
本文讨论András Sárközy 2001年未解决问题集中的第66、67题，证明了素数稀疏子集和集的最优结果。
This note simplifies the proof of a recent result on the oscillation of the prime product in Martens Theorem, and provides a quantitative expression for the error term. In addition, the corresponding oscillation results for the finite sums of the reciprocal of the primes up to a fixed number and other finite sums and products are also given.
In this note, we characterize all functions $f : \mathbb{N} \rightarrow \mathbb{C}$ such that $f(x_1^2+ \cdots + x_k^2)=f(x_1)^2+ \cdots + f(x_k)^2$, where $k \geq 3$ and $x_1, \cdots, x_k$ are positive integers.
It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.
In a recent note W. Kohnen asks whether the values of Dedekind sums are dense in the field of $p$-adic numbers. The present paper answers this question. Dedekind sums do not approximate units of $\mathbb Z_2$ or $\mathbb Z_3$, so they are not dense in $\mathbb Q_2$ or $\mathbb Q_3$. But they are dense in $\mathbb Q_p$ if $p\ge 5$.
A note on new type degenerate Srirling numbers of the first kind
本研究引入新型无符号退化第一类斯特林数，借鉴Adell-Lekuona的方法给出其含随机变量期望及经典第一类斯特林数的有限和表示，还定义了对应的新型退化第二类斯特林数并推导其含第二类斯特林数的有限和表达式。
In this note we consider the theorem established in arXiv:1912.07171 concerning the sums of powers of the first $n$ positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and show that it can be used to demonstrate the classical theorem of Faulhaber for both cases of odd and even $k$.
We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were systematically first studied by Berndt and Zaharescu; their proofs involved complex contour integration. We show how to prove most of Berndt-Zaharescu's and some new identities by elementary methods of discrete Fourier Analysis.
Notes on certain binomial harmonic sums of Sun's type
本文证明并推广了Sun关于包含调和数和二项式系数乘积的无穷级数的猜想，通过自守对象方法求得闭合形式。
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 $x\ne y$.
In this note we prove a new estimate on so-called GCD sums (also called Gál sums), which, for certain coefficients, improves significantly over the general bound due to de la Bretèche and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher, and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.
We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} χ(n)$, where $χ$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty$ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty$, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $χ$, even when $H = q/\log^{A}q$ for any fixed $A > 0$. On the other hand, we show it is true…
In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.
Multiple zeta values (MZVs) with certain repeated arguments or certain sums of cyclically generated MZVs are evaluated as rational multiple of powers of $π^2$. In this paper, we give a short and simple proof of the remarkable evaluations of MZVs established by D. Borman and D. M. Bradley.
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In the present note, we prove new lower bounds on large values of character sums $Δ(x,q):=\max_{χ\neq χ_0} \vert \sum_{n\leq x} χ(n)\vert$ in certain ranges of $x$. Employing an implementation of the resonance method developed in a work involving the author in order to exhibit large values of $L$- functions, we improve some results of Hough in the range $\log x = o(\sqrt{\log q})$. Our results are expressed using the counting function of $y$- friable integers less than $x$ where we improve the level of smoothness $y$ for short intervals.
We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya and Vinogradov inequality. Our argument is based on getting an upper bound for the l1 norm of the Fourier coefficients of a generalized arithmetic progression. Our method also applies to give upper bounds for polynomial exponential sums.
In this paper we consider a variety of mixed character sums. In particular we extend a bound of Heath-Brown and Pierce to the case of squarefree modulus, improve on a result of Chang for mixed sums in finite fields, we show in certain circumstances we may improve on some results of Pierce for multidimensional mixed sums and we extend a bound for character sums with products of linear forms to the setting of mixed sums.
We improve a recent result by Shparlinski and Banks related to sums with convolution of Dirichlet characters.
Structure of large quadratic character sums
本文基于Lamzouri的工作，研究大二次特征和的分布，并获得了具有大特征和的二次特征的结构结果。
Dedekind sums $s(m,n)$ occur in many fields of mathematics. Since $s(m_1,n)=s(m_2,n)$ if $m_1\equiv m_2$ mod $n$, it is natural to ask which of the Dedekind sums $s(m,n)$, $0\le m<n$, take equal values. So far no simple criterion is known by which the equality of $s(m_1,n)$ and $s(m_2,n)$ could be decided. In this note we show how to obtain non-obvious examples of equal Dedekind sums. We consider two cases which mark the extreme possibilities for the argument $n$, namely, $n$ a prime power and $n$ square-free. Whereas we can give a partial overview of equal Dedekind sums in the prime power cas…
Let $q$ be a positive integer, $χ$ a nontrivial character mod $q$, $\mathcal{I}$ an interval of length not exceeding $q.$ In this paper we shall study the character sum analogue of the well-known Kloosterman sum,\[\sum_{\substack{a\in\mathcal{I} \gcd(a,q)=1}}χ(ma+n\overline{a}),\] where $\overline{a}$ is the multiplicative inverse of $a\bmod q$. The mean square values and bilinear forms for such sums are proved.
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