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 for "almost all" characters on the range $q^{1-o(1)} \leq H = o(q)$.
Using Pólya's Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.
Using a probabilistic model, based on random walks on the additive group $\mathbb{Z}/m\mathbb{Z}$, we prove that the values of certain real character sums are uniformly distributed in residue classes modulo $m$.
Let $χ$ be a non-real Dirichlet character modulo a prime $q$. In this paper we prove that the distribution of the short character sum $S_{χ,H}(x)=\sum_{x< n\leq x+H} χ(n)$, as $x$ runs over the positive integers below $q$, converges to a two-dimensional Gaussian distribution on the complex plane, provided that $\log H=o(\log q)$ and $H\to\infty$ as $q\to\infty$. Furthermore, we use a method of Selberg to give an upper bound on the rate of convergence.
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.
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.
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.
We consider character sums determined by isogenies of elliptic curves over finite fields. We prove a congruence condition for character sums attached to arbitrary cyclic isogenies, and produce explicit formulas for isogenies of small degree.
We obtain a Burgess-type bound for character sums over unions of intervals. The result follows from the argument of Heath-Brown, with an improvement in one of the steps.
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.
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.
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.
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 $1$).
We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.
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）
A classical result of Paley shows that there are infinitely many quadratic characters $χ\mod{q}$ whose character sums get as large as $\sqrt{q}\log \log q$; this implies that a conditional upper bound of Montgomery and Vaughan cannot be improved. In this paper, we derive analogous lower bounds on character sums for characters of odd order, which are best possible in view of the corresponding conditional upper bounds recently obtained by the first author.
We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the multiplicative characters modulo a prime $p$ we break the barrier of $p^{1/4}$ for ranges of individual variables.
We study smoothed character sums involving $\sum_{m,n} ( \frac{m}{n} )_2$, where $( \frac{m}{n} )_2$ denotes the quadratic symbol in the Gaussian field. We extend previously known results to obtain asymptotic formulas for the sums considered to larger ranges of $m$ and $n$.
We improve a recent result of B. Hanson (2015) on multiplicative character sums with expressions of the type $a + b +cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a + b(c+d)$. These bounds rely on some recent advances in additive combinatorics.
In this paper, we prove that for any $A>0$ there exist infinitely many primes $p$ for which sums of the Legendre symbol modulo $p$ over an interval of length $(\ln p)^A$ can take large values.
A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis.
We use the large sieve inequality for smooth numbers due to S. Drappeau, A. Granville and X. Shao (2017), together with some other arguments, to improve their bounds on the frequency of pairs $(q,χ)$ of moduli $q$ and primitive characters $χ$ modulo $q$, for which the corresponding character sums with smooth numbers are large.
We consider the size of large character sums, proving new lower bounds for the quantity $Δ(N,q) = \sup_{χ\neq χ_0 mod q} |\sum_{n < N} χ(n)|$ for almost all ranges of $N$. The results are proven using the resonance method and saddle point analysis.
We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence $\{[αn + β] : n =1,2,... \}$, where $α,β\in\R$, and $α$ is irrational. Our bounds are nontrivial over the same short intervals for which the classical character sum estimates of Burgess have been established.
The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This leads to a power-saving if the modulus is smooth enough. As an application, a fast, and potentially practical, method to compute Dirichlet L-functions with complexity exponent 1/3 for smooth enough moduli is derived.
We show that the binomial and related multiplicative character sums $$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} χ(x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} χ_1 (x)χ_2(Ax^k +B), $$ have a simple evaluation for large enough $m$ (for $m\geq 2$ if $p\nmid ABk$).
For integer $q$, let $χ$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}Λ(n)χ(n+a),$$ where $Λ(n)$ is the von Mangoldt function. This bound improves and extends the range of a result of Friedlander, Gong and Shparlinski
We prove that the existence of exceptional real zeroes of Dirichlet $L$-functions would lead to cancellations in the sum $\sum_{p\leq x} \Kl(1, p)$ of Kloosterman sums over primes, and also to sign changes of $\Kl(1, n)$, where $n$ runs over integers with exactly two prime factors. Our arguments involve a variant of Bombieri's sieve, bounds for twisted sums of Kloosterman sums, and work of Fouvry and Michel on sums of $\left| \Kl(1, n)\right|$.
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 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 show that for any mod $2^m$ characters, $χ_1, χ_2,$ the complete exponential sum, $$ \sum_{x=1}^{2^m}χ_1(x) χ_2(Ax^k+B), $$ has a simple explicit evaluation.
We use the $q$-analogue of van der Corput's method to estimate short character sums to smooth moduli. If $χ$ is a primitive Dirichlet character modulo a squarefree, $q^δ$-smooth integer $q$ we show that $$L(\frac12,χ)\ll_εq^{\frac{27}{164}+O(δ)+ε}.$$
We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U} \sum_{v=1}^V α_u β_v \mathbf{\,e}_p(u/f(v)) $$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$ elements, with integers $U$, $V$, a polynomial $f\in {\mathbb F}_p[X] $ and some complex weights $\{α_u\}$, $\{β_v\}$. In particular, for $f(X)=aX+b$ we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bounds for similar sums with multiplicative characters of ${\mathbb F}_p$.
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.
Soochow University（苏州大学）；Hainan Bielefeld University of Applied Sciences（海南比勒费尔德应用科技大学）；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）；Tongji University（同济大学）
We combine a classical idea of Postnikov (1956) with the method of Korobov (1974) for estimating double Weyl sums, deriving new bounds on short character sums when the modulus $q$ has a small core $\prod_{p\mid q}p$. Using this estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for the corresponding $L$-functions. In turn, this allows us to improve the error term in the asymptotic formula for primes in short arithmetic progressions modulo a power of a fixed prime. As yet another application of our bounds, we substantially extend the region free of Siegel zeros.
In their recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, these results are used to bound sums of the Mobius function twisted by characters of the same type, complementing and improving some earlier work of B. Green (2012). To achieve this, we obtain a series of results about the size and zero-free region of $L$-functions with the same class of moduli.
Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n <= x is o(x); they show that this holds if log x / log log q -> infinity and q -> infinity (q is the size of the finite field).
In this paper, we consider the distribution of the continuous paths of Dirichlet character sums modulo prime $q$ on the complex plane. We also find a limiting distribution as $q \rightarrow \infty$ using Steinhaus random multiplicative functions, stating properties of this random process. This is motivated by Kowalski and Sawin's work on Kloosterman paths.
Let $\mathcal L^+$ be the set of all primes $p$ for which the sums of $\left(\frac{n}{p}\right)$ over the interval $[1,N]$ are non-negative for all $N$. We prove that the estimate \[ |\mathcal L^+\cap [1,x]|\ll \frac{x}{\ln x(\ln\ln x)^{c-o(1)}} \] holds for $c\approx 0.0368$
We prove new bounds for sums of multiplicative characters over sums of set with small doubling and applying this result we break the square--root barrier in a problem of Balog concerning products of differences in a field of prime order.
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.
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…
An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then rivisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes $p$ of the form $6k+1$ by a binary quadratic form in integers of a subfield of the cyclotomic field of the $p$-th roots of unity.
