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 dimensions.
In this work we establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of "admissible" forms. This $n$-dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least $q^β$, with $β> 1/2 - 1/(2(n+1))$. This is the first Burgess bound that applies in all dimensions to generic forms of arbitrary degree. Our approach capitalizes on a recent stratification result for complete multiplicative character sums evaluated at rational functions, due to the s…
In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski. In particular, for any nontrivial multiplicative character $χ$ modulo a prime $q$ and any integer $r\ge 2$, we show that $$ \sum_{M<n\le M+N}χ(n) = O\left( N^{1-1/r}q^{(r+1)/4r^2}(\log q)^{1/4r}\right), $$ which sharpens previous results by a factor $(\log q)^{1/4r}$. Our improvement comes from averaging over numbers with no small prime factors rather than o…
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.
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.
In recent years the question of maximizing GCD sums regained interest due to its firm link with large values of $L$-functions. In the present paper we initiate the study of minimizing for positive weights~$w$ of normalized $L^1$- norm the sum $\sum_{m_1 , m_2 \leqslant N} w({m_1})w({m_2})\frac{(m_1,m_2)}{\sqrt{m_1m_2}} $. We consider as well the intertwined question of minimizing a weighted version of the usual multiplicative energy. We give three applications of our results. Firstly we obtain a logarithmic refinement of Burgess' bound on character sums $\displaystyle{\sum_{M<n\leqslant M+N}χ(…
Let $χ$ be a Dirichlet character modulo $p$, a prime. In applications, one often needs estimates for short sums involving $χ$. One such estimate is the family of bounds known as \emph{Burgess' bound}. In this paper, we explore several minor adjustments one can make to the work of Enrique Treviño on explicit versions of Burgess' bound. For an application, we investigate the problem of the existence of a $k$th power non-residue modulo $p$ which is less than $p^α$ for several fixed $α$. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields.
Let $g$ be a fixed Hecke cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and $χ$ be a primitive Dirichlet character of conductor $M$. The best known subconvex bound for $L(1/2,g\otimes χ)$ is of Burgess strength. The bound was proved by a couple of methods: shifted convolution sums and the Petersson/Kuznetsov formula analysis. It is natural to ask what inputs are really needed to prove a Burgess-type bound on $\rm GL(2)$. In this paper, we give a new proof of the Burgess-type bounds ${L(1/2,g\otimes χ)\ll_{g,\varepsilon} M^{1/2-1/8+\varepsilon}}$ and $L(1/2,χ)\ll_{\varepsilon} M^{1/4-1/16+\varepsilo…
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.
In this paper we consider the problem of estimating character sums to composite modulus and obtain some progress towards removing the cubefree restriction in the Burgess bound. Our approach is to estimate high order moments of character sums in terms of solutions to congruences with Kloosterman fractions and we deal with this problem by extending some techniques of Bourgain, Garaev, Konyagin and Shparlinski and Bourgain and Garaev from the setting of prime modulus to composite modulus. As an application of our result we improve an estimate of Norton.
We estimate weighted character sums with determinants $ad-bc $ of $2\times 2$ matrices modulo a prime $p$ with entries $a,b,c,d $ varying over the interval $ [1,N]$. Our goal is to obtain nontrivial bounds for values of $N$ as small as possible. In particular, we achieve this goal, with a power saving, for $N \ge p^{1/8+\varepsilon}\ $ with any fixed $\varepsilon>0$, which is very likely to be the best possible unless the celebrated Burgess bound is improved. By other techniques, we also treat more general sums but sometimes for larger values of $N$.
We generalize our previous method on subconvexity problem for $\mathrm{GL}_2 \times \mathrm{GL}_1$ with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, i.e., the bound $|L(1/2,χ)| \ll_{\mathbf{F},ε} \mathbf{C}(χ)^{1/4-(1-2θ)/16+ε}$ for varying Hecke characters $χ$ over a number field $\mathbf{F}$ with analytic conductor $\mathbf{C}(χ)$. As a main tool, we apply the extended theory of regularized integral due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.
In recent years, maximizing Gál sums regained interest due to a firm link with large values of $L$-functions. In the present paper, we initiate an investigation of small sums of Gál type, with respect to the $L^1$-norm. We also consider the intertwined question of minimizing weighted versions of the usual multiplicative energy. We apply our estimates to: (i) a logarithmic refinement of Burgess' bound on character sums, improving previous results of Kerr, Shparlinski and Yau; (ii) an improvement on earlier lower bounds by Louboutin and the second author for the number of non vanishing theta fun…
In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for primitive characters. Given a primitive character $χ$ modulo $q$, we prove the following upper bound \begin{align*} \left| \sum_{1 \le n\le N} χ(n) \right|\le c \sqrt{q} \log q, \end{align*} where $c=3/(4π^2)+o_q(1)$ for even characters and $c=3/(8π)+o_q(1)$ for odd characters, with explicit $o_q(1)$ terms. This improves a result of Frolenkov and Soundararajan for large $q$. We proceed, following Hildebrand, obtaining the explicit version of a result by Montgomery--Vaughan on partial Gaussian sums and…
ADA University（阿塞拜疆科技大学）；Northwestern University（西北大学）；Universidad Autónoma de Santo Domingo（圣多明各自治大学）；University of British Columbia（不列颠哥伦比亚大学）；Lake Forest College（湖森林学院）
Given a Dirichlet character $χ$ modulo $q$ and its associated $L$-function, $L(s,χ)$, we provide an explicit version of Burgess' estimate for $|L(s, χ)|$. We use partial summation to provide bounds along the vertical lines $\Re{s} = 1 - {r}^{-1}$, where $r$ is a parameter associated with Burgess' character sum estimate. These bounds are then connected across the critical strip using the Phragmén--Lindelöf principle. In particular, for $σ\in [\frac{1}{2}, \frac{9}{10}]$, we establish $$|L(σ+ it, χ)| \leq (1.105) (0.692)^σq^{\frac{31}{80}-\frac{2}{5}σ}(\log{q})^{\frac{33}{16}-\frac{9}{8}σ} |σ+ i…
Large sieve inequality for sums of Legendre symbols over short intervals
Université Jean Monnet, Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, ICJ UMR5208（让·莫内大学、里昂中央理工学院、里昂国立应用科学学院、里昂第一大学、法国国家科学研究中心、ICJ UMR5208）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, University of Bristol（布里斯托大学数学系）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
Let $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary number field, $(p)=\mathfrak{p}\mathfrak{p}'$ be a split odd prime and $ψ$ be a Hecke character of conductor $\mathfrak{p}$. Let $L(s,ψ)$ be the associated $L$-function. We prove the Burgess bound in $t$-aspect and a hybrid bound in conductor aspect, \begin{equation*} L(1/2+it,ψ)\ll_{D,\varepsilon} (1+|t|)^{3/8+\varepsilon}p^{1/8} \end{equation*} for $p\ll t$. In Appendix A, we present the ideas for an elementary proof of Voronoi summation formula for holomorphic cusp forms with CM and squarefree level. This is done by exploiting the lattice struct…
We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.
We consider a family of character sums as multiplicative analogues of Kloosterman sums. Using Gauss sums, Jacobi sums and Deligne's bound for hyper-Kloosterman sums, we establish asymptotic formulae for any real (positive) moments of the above character sum as the character runs over all non-trivial multiplicative characters mod $p.$ Moreover, an arcsine law is also established as a consequence of the method of moments. The evaluations of these moments also allow us to obtain asymptotic formulae for moments of such character sums weighted by special $L$-values (at $1/2$ and $1$).
In this paper, we prove a lower bound for $\underset{χ\neq χ_0}{\max}\bigg|\sum_{n\leq x} χ(n)\bigg|$, when $x= \frac{q}{(\log q)^B}$. This improves on a result of Granville and Soundararajan for large character sums when the range of summation is wide. When $B$ goes to zero, our lower bound recovers the expected maximal value of character sums for most characters.
Large values of quadratic character sums revisited
Tongji University（同济大学）；Hainan Bielefeld University of Applied Sciences（海南比勒费尔德应用科学大学）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；Hunan University（湖南大学）
