A Pólya--Vinogradov inequality for short character sums

On the constant in the Polya-Vinogradov inequality

In this paper we obtain a new constant in the Pólya-Vinogradov inequality. Our argument follows previously established techniques which use the Fourier expansion of an interval to reduce to Gauss sums. Our improvement comes from approximating an interval by a function with slower decay on the edges and this allows for a better estimate of the $\ell_1$ norm of the Fourier transform. This approximation induces an error for our original sums which we deal with by combining some ideas of Hildebrand with Garaev and Karatsuba concerning long character sums.

关于素数乘积上指数和的Bombieri- Vinogradov定理

A Bombieri-Vinogradov theorem for exponential sums over products of k primes

本文提出了一种关于素数乘积上指数和的Bombieri-Vinogradov定理，并给出了其在数论中的应用及下界估计。

Improving the Burgess bound via Polya-Vinogradov

We show that even mild improvements of the Polya-Vinogradov inequality would imply significant improvements of Burgess' bound on character sums. Our main ingredients are a lower bound on certain types of character sums (coming from works of the second author joint with J. Bober and Y. Lamzouri) and a quantitative relationship between the mean and the logarithmic mean of a completely multiplicative function.

Pólya-Vinogradov and the least quadratic nonresidue

It is well-known that cancellation in short character sums (e.g. Burgess' estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess' work, so it is desirable to find a new approach to nonresidues. The goal of this note is to demonstrate a new line of attack via long character sums, a currently active area of research. Among other results, we demonstrate that improving the constant in the Pólya-Vinogradov inequality would lead to significant progress on nonresidues. Moreover, conditionally on a conjecture on long character sum…

Multiplicative mimicry and improvements of the Polya-Vinogradov inequality

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd order character sums. Furthermore, conditionally on the Generalized Riemann Hypothesis we obtain a bound for odd order character sums which is best possible.

Multigrade efficient congruencing and Vinogradov's mean value theorem

We develop a multigrade enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, thereby obtaining near-optimal estimates for $\tfrac{5}{8}k^2<s\le k^2-k+1$. There are numerous applications. In particular, when $k$ is large, the anticipated asymptotic formula in Waring's problem is established for sums of $s$ $k$th powers of natural numbers whenever $s>1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.

Incomplete Gauss sums modulo primes

We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov's mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.

Explicit Improvements to the Burgess Bound Via Pólya-Vinogradov

We make explicit a theorem of Fromm and Goldmakher [arXiv:1706.03002], which states that one can improve Burgess' bound for short character sums simply by improving the leading constant in the Pólya-Vinogradov inequality. Towards achieving this, we establish explicit versions of several estimates related to the mean values of real multiplicative functions and the Dickman function.

Vinogradov's mean value theorem via efficient congruencing

We obtain estimates for Vinogradov's integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring's problem holds for sums of s kth powers of natural numbers whenever s is at least 2k^2+2k-3.

Large odd order character sums and improvements of the Pólya-Vinogradov inequality

For a primitive Dirichlet character $χ$ modulo $q$, we define $M(χ)=\max_{t } |\sum_{n \leq t} χ(n)|$. In this paper, we study this quantity for characters of a fixed odd order $g\geq 3$. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character $χ$ we have $$M(χ)\ll_{\varepsilon} \sqrt{q}(\log q)^{1-δ_g}(\log\log q)^{-1/4+\varepsilon},$$ where $δ_g := 1-\frac{g}π\sin(π/g)$. This improves upon the works of Granville and Soundararajan and of Goldmakher. Furthermore, assuming the Generalized Ri…

Singular Gauss sums, Polya-Vinogradov inequality for $GL(2)$ and growth of primitive elements

We establish an analogue of the classical Polya-Vinogradov inequality for $GL(2, \F_p)$, where $p$ is a prime. In the process, we compute the `singular' Gauss sums for $GL(2, \F_p)$. As an application, we show that the collection of elements in $GL(2,\Z)$ whose reduction modulo $p$ are of maximal order in $GL(2, \F_p)$ and whose matrix entries are bounded by $x$ has the expected size as soon as $x\gg p^{1/2+\ep}$ for any $\ep>0$. In particular, there exist elements in $GL(2,\Z)$ with matrix entries that are of the order $O(p^{1/2+\ep})$ whose reduction modulo $p$ are primitive elements.

Burgess bounds for short mixed character sums

This paper proves nontrivial bounds for short mixed character sums by introducing estimates for Vinogradov's mean value theorem into a version of the Burgess method.

Solution-free sets for sums of binary forms

We obtain quantitative estimates for the asymptotic density of subsets of the two-dimensional integer lattice which contain only trivial solutions to an additive equation involving binary forms. In the process we develop an analogue of Vinogradov's mean value theorem applicable to binary forms.

Character sums to smooth moduli are small

Recently, Granville and Soundararajan have made fundamental breakthroughs in the study of character sums. Building on their work and using estimates on short character sums developed by Graham-Ringrose and Iwaniec, we improve the Polya-Vinogradov inequality for characters with smooth conductor.

A note on medium and short character sums

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.

Character sums over Bohr sets

We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to classical character sum bounds of Polya-Vinogradov and Burgess. These estimates are applied to obtain results on recurrence mod $p$ by special elements.

On Character Sums and Exponential Sums over Generalized Arithmetic Progressions

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.

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。

Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed.

Three conjectures about character sums

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya-Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess' estimate for short character sums, and upper bounds for $L(1,χ)$ and $L(1+it,χ)$) are more-or-less "equivalent". We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.

An elementary approach to character sums over multiplicative subgroups

Let $p$ be an odd prime. Using I. M. Vinogradov's bilinear estimate, we present an elementary approach to estimate nontrivially the character sum $$ \sum_{x\in H}χ(x+a),\qquad a\in\Bbb F_p^*, $$ where $H<\Bbb F_p^*$ is a multiplicative subgroup in finite prime field $\Bbb F_p$. Some interesting mean-value estimates are also provided.

Short character sums for composite moduli

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square free moduli and also on the result due to Gallagher and Iwaniec when the core $q'=\prod_{p|q}p$ of the modulus $q$ satisfies $\log q'\sim \log q$. Some applications to zero free regions of Dirichlet L-functions and the $\rm{P\acute{o}lya}$ and Vinogradov inequalities are indicated.

Optimal mean value estimates for the quadratic Weyl sums in two dimensions

We use decoupling theory to prove a sharp (up to $N^ε$ losses) estimate for Vinogradov's mean value theorem in two dimensions

Mean value estimates for Weyl sums in two dimensions

We use decoupling theory to estimate the number of solutions for quadratic and cubic Parsell--Vinogradov systems in two dimensions.

The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture

For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\eps)$ for every fixed $\eps>0$. This conjecture follows from the generalised Riemann hypothesis, and is known to hold for almost all primes $p$ but remains open in general. In this paper we show that Vinogradov's conjecture also follows from the Elliott-Halberstam conjecture on the distribution of primes in arithmetic progressions, thus providing a potential "non-multiplicative" route to the Vinogradov conjecture. We also give a variant of this argument that obtains bounds …

Weyl sums over integers with digital restrictions

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the "main conjecture" on the Vinogradov mean value theorem which has been established by Bourgain, Demeter and Guth (2016) as well as by Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets that are defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.

The sliding-sum method for short exponential sums

We introduce a method to estimate sums of oscillating functions on finite abelian groups over intervals or (generalized) arithmetic progressions, when the size of the interval is such that the completing techniques of Fourier analysis are barely insufficient to obtain non-trivial results. In particular, we prove various estimates for exponential sums over intervals in finite fields and related sums just below the Polya-Vinogradov range, and derive applications to equidistribution problems.

Burgess bounds for short character sums evaluated at forms II: the mixed case

This work proves a Burgess bound for short mixed character sums in $n$ dimensions. The non-principal multiplicative character of prime conductor $q$ may be evaluated at any "admissible" form, and the additive character may be evaluated at any real-valued polynomial. The resulting upper bound for the mixed character sum is nontrivial when the length of the sum is at least $q^β$ with $β> 1/2 - 1/(2(n+1))$ in each coordinate. This work capitalizes on the recent stratification of multiplicative character sums due to Xu, and the resolution of the Vinogradov Mean Value Theorem in arbitrary dimension…

一个扩展的维诺格拉多夫均值定理

An extended Vinogradov's mean value theorem

本文通过圆法与变量平移论证，为扩展的维诺格拉多夫均值定理相关指数和建立新均值估计，在d=2,3时获得尖锐上界，d≥4时依赖去耦不等式获得类似结果。

On short sums of trace functions

We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the "Polyá-Vinogradov gap" in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform, and we give applications related to trace functions over finite fields.

Burgess bounds for multi-dimensional short mixed character sums

This paper proves Burgess bounds for short mixed character sums in multi-dimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial, and a product of multiplicative Dirichlet characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation-dilation invariant systems in order to prove character sum bounds in any dimension that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding the given polynomial into an adv…

Distribution of short subsequences of inversive congruential pseudorandom numbers modulo $2^t$

In this paper we study the distribution of very short sequences of inversive congruential pseudorandom numbers modulo $2^t$. We derive a new bound on exponential sums with such sequences and use it to give estimate their discrepancy. The technique we use, based the method of N. M. Korobov (1972) of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford (2002), has never been used in this area and is very likely to find further applications.

素数上的指数和

Exponential sums over primes

该研究改进了1937年Vinogradov关于素数上指数和的经典界，得到了含N^(19/24)的新上界，相关成果属于数论领域。

无大素因子整数上的指数和

Exponential sums over integers without large prime divisors

本文通过更精确地利用无大素因子整数的因数分解并结合Type I和Type II界，改进了此类整数及其 $ν$ 次幂上的指数和的现有界限。

任意集合上带迹函数的双线性形式及其在 Sato-Tate 中的应用

Bilinear forms with trace functions over arbitrary sets, and applications to Sato-Tate

本文证明了丰饶层迹函数在任意子集支撑下的一般双线性形式非平凡上界，结合加性组合与Kloosterman层乘积和结果突破了Pólya-Vinogradov障碍，并推导出稀疏族中Kloosterman和与椭圆曲线Frobenius迹的Sato-Tate分布。

Averages of exponential twists of the von Mangoldt function

In this paper, we obtain some improved results for the exponential sum $\sum_{x<n\leq 2x}Λ(n)e(αk n^θ)$ with $θ\in(0,5/12),$ where $Λ(n)$ is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Vinogradov \cite{Va} and Murty-Srinivas \cite{Mu}.

Restricted mean value theorems and metric theory of restricted Weyl sums

We study an apparently new question about the behaviour of Weyl sums on a subset $\mathcal{X}\subseteq [0,1)^d$ with a natural measure $μ$ on $\mathcal{X}$. For certain measure spaces $(\mathcal{X}, μ)$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for $μ$-almost all points of $\mathcal{X}$ the Weyl sums satisfy the square root cancellation law. Moreover we characterise the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $(\mathcal{X}, μ)$. We obtain general res…

双线性形式与迹函数

Bilinear forms with trace functions

本文提出了一种基于分层定理和鲁棒准则的新方法，用于在波利亚-文纳戈夫范围内获得双线性迹函数和的非平凡界。

On the size of the maximum of incomplete Kloosterman sums

Let $t:\mathbb{F}_{p}\rightarrow\mathbb{C}$ be a complex valued function on $\mathbb{F}_{p}$. A classical problem in analytic number theory is to bound the maximum of the absolute value of the incomplete sum \[ M(t):=\max_{0\leq H<p}\Big|\frac{1}{\sqrt{p}}\sum_{0\leq n < H}t(n)\Big|. \] In this very general context one of the most important results is the Pólya-Vinogradov bound \[ M(t)\leq \left\|K\right\|_{\infty}\log 3p. \] where $K:\mathbb{F}_{p}\rightarrow\mathbb{C}$ is the normalized Fourier transform of $t$. In this paper we provide a lower bound for incomplete Kloosterman sum, namely we…

椭圆集上的高效同余方法：二次情形

Efficient congruencing in ellipsephic sets: the quadratic case

本文通过数字限制下的高效同余方法，改进了二次Vinogradov方程组解数的上界，利用特定数字集合的稀疏表示性质获得更优结果。

An asymptotic for sums of Lyapunov exponents in families

Let f_t be a meromorphic family of endomorphisms of P^N_C of degree at least 2, and let L(f_t) be the sum of Lyapunov exponents associated to f_t. Favre showed that L(f_t)=L(f)\log|t^{-1}|+o(\log|t^{-1}|) as t -> 0, where L(f) is the sum of Lyapunov exponents on the generic fibre, interpreted as an endomorphism of some projective Berkovich space. Under some additional constraints on the family, we provide an explicit error term.

On a Hybrid Version of the Vinogradov Mean Value Theorem

Given a family $φ= (φ_1, \ldots, φ_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $ρ$, we consider the mean value $$ M_{k, ρ} (φ, N) = \int_{\mathbf{x} \in [0,1]^k} \sup_{\mathbf{y} \in [0,1]^{d-k}} \left| S_φ(\mathbf{x}, \mathbf{y}; N) \right|^ρd\mathbf{x} $$ of exponential sums $$ S_φ( \mathbf{x}, \mathbf{y}; N) = \sum_{n=1}^{N} \exp\left(2 πi \left(\sum_{j=1}^k x_j φ_j(n)+ \sum_{j=1}^{d-k}y_jφ_{k+j}(n)\right)\right), $$ where $\mathbf{x} = (x_1, \ldots, x_k)$ and $\mathbf{y} =(y_1, \ldots, y_{d-k})$. The case of poly…

广义除数函数的三重卷积和及相关素数上的和

Triple convolution sums of the generalised divisor functions and related sums over primes

本文研究广义除数函数的三重卷积和及其在素数上的应用，通过多变量Dirichlet级数理论和概率论方法推导出渐进行为及下界。

On the Waring--Goldbach problem for eighth and higher powers

Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem. We obtain new results for all exponents $k\ge 8$, and in particular establish that $H(k)\le (4k-2)\log k+k-7$ when $k$ is large, giving the first improvement on the classical result of Hua from the 1940s.

由加性函数加权的指数和

Exponential sums weighted by additive functions

本文研究了由加性函数加权的指数和，证明了其紧界，并应用Hardy-Littlewood圆法解决了与总素因子个数相关的Goldbach-Vinogradov三元问题的渐进行为。

Small fractional parts of binary forms

We obtain bounds on fractional parts of binary forms of the shape $$Ψ(x,y)=α_k x^k+α_l x^ly^{k-l}+α_{l-1}x^{l-1}y^{k-l+1}+\cdots+α_0 y^k$$ with $α_k,α_l,\ldots,α_0\in\mathbb{R}$ and $l\leq k-2.$ By exploiting recent progress on Vinogradov's mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent $σ$, depending on $k$ and $l,$ such that \begin{equation*} \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|Ψ(x,y)\|\leq X^{-σ+ε}.\end{equation*}

素数上的短指数和之和

Sum of short exponential sums with prime numbers

本文针对素数上的短指数和，在更一般条件下得到了一个加强并推广维诺格拉多夫估计的上界，改进了相关和的阶。

高阶特征的大和

Large Sums of High Order Characters

本文研究模素数高阶本原特征的短和与取值分布，利用跨幂次平均的二阶矩估计及矫饰数论、加性组合学方法，证明短区间内特征值不会聚集于1，并平均改进Pólya–Vinogradov不等式。

奇数阶狄利克雷特征的最大和的尖锐界

Sharp bounds for maximal sums of odd order Dirichlet characters

本文在GRH下改进奇数阶狄利克雷特征最大和的Pólya-Vinogradov上界，并证明其尖锐性，优于以往结果。

短特征和的变号与Fekete多项式的实零点

Sign changes of short character sums and real zeros of Fekete polynomials

本文提出一种量化加权乘性函数和变号次数的通用方法，证明了典型基本判别式下实特征部分和与Fekete多项式实零点数的下界，并依赖Dirichlet L函数的联合分布与集中度估计，进一步给出了零点数的非平凡上界及无远离1零点的判别式族。

高阶特征的大和 II

Large sums of high order characters II

本文研究模 q 本原特征在大阶情形下水平集 χ(n)=α 的计数问题，通过改进短特征和的均方估计，在 x>q^δ 时得到非平凡上界，并改进 Pólya-Vinogradov 不等式。

关于一个具有形式 $\mathbf{p=x^2+y^2+1}$ 的素数的三元丢番图不等式

A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$

本文证明对任意 $1<c<427/400$，三元 Piatetski-Shapiro 不等式在素数中有解，且其中一个素数可表示为 $x^2+y^2+1$，为此建立了新的 Bombieri--Vinogradov 型指数和结果。

关于破弧上外尔和的积分中值定理

An Integral Mean Value Theorem for Weyl Sums over Broken Arcs

本文针对破弧上的外尔和，通过丢番图逼近、Vinogradov主值定理及改进的外尔差分法，突破幂的奇偶限制得到积分中值的新估计，完成相关证明。

模半整权L-函数的扭曲二阶矩

The twisted second moment of modular half integral weight $L$--functions

本文无条件证明了半整权 Kohnen 新形式 L 函数在大素数模特征下二阶矩的渐近公式，通过谱分析和 Salié 和估计给出幂节省误差项，从而得到无 Euler 乘积 Dirichlet 级数的平均 Lindelöf 结果。