Density of non-residues in Burgess-type intervals and applications
W. D. Banks, M. Z. Garaev, D. R. Heath-Brown, I. E. Shparlinski
备注：
In the new version we use an idea of Roger Heath-Brown (who is now a co-author) to simply the proof and improve the main results of the previous version, 14 pages
We show that for any fixed $\eps>0$, there are numbers $δ>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $δN$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.
We give an explicit version of a result due to D. Burgess. Let $χ$ be a non-principal Dirichlet character modulo a prime $p$. We show that the maximum number of consecutive integers for which $χ$ takes on a particular value is less than $\left\{\frac{πe\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$, where the $o(1)$ term is given explicitly.
The Burgess inequality is the best upper bound we have for the character sum $S_χ(M,N) = \sum_{M<n\le M+N} χ(n).$ Until recently, no explicit estimates had been given for the inequality. In 2006, Booker gave an explicit estimate for quadratic characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than $10^{140}$. Both of their explicit estimates are on restricted ranges. In this paper we prove an explicit estimate that works for any $M$ and $N$. We…
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.
For any prime $p$, let $y(p)$ denote the smallest integer $y$ such that every reduced residue class $\pmod p$ is represented by the product of some subset of $\{1,\dots,y\}$. It is easy to see that $y(p)$ is at least as large as the smallest quadratic nonresidue $\pmod p$; we prove that $y(p) \ll_\varepsilon p^{1/(4 \sqrt e)+\varepsilon}$, thus strengthening Burgess's classical result. This result is of intermediate strength between two other results, namely Burthe's proof that the multiplicative group $\pmod p$ is generated by the integers up to $O_\varepsilon(p^{1/(4 \sqrt e)+\varepsilon}$, …
We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+ψ(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $ψ(Q)\to\infty$ as $Q\to\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.
We show that for any $\varepsilon > 0$ and a sufficiently large cube-free $q$, any reduced residue class modulo $q$ can be represented as a product of $14$ integers from the interval $[1, q^{1/4e^{1/2} + \varepsilon}]$. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications to Fermat quotients.
We consider the problem of bounding the dimension of Hilbert cubes in a finite field $\mathbb{F_p}$ that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is $O_ε(p^{1/8+ε})$ for any $ε> 0$, matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.
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.
We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial ingredient in our work are recent equidistribution results of N. Katz.
Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number formula for the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$.
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）
In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv5\pmod8$, then $$\prod_{0<x<p/2,(\frac{x}{p})=1}x\equiv(-1)^{1+r}\pmod p,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelberger's congruence and values of Dirichlet L-series at negative integers.
We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),
We prove that the sequence $n!\,(\bmod\,p)$ occupies at least $\sqrt{\frac{3}{2}N}$ residue classes in the short interval $H\le n \le H+N$ and $N\gg p^{\frac{1}{4}}$ improving previously known trivial bound $\sqrt{N}.$ In the other direction, we estimate the average number of residue classes missed by the sequence $n!\,(\bmod\,p)$ for $p\le x.$
Given collections A and B of residue classes modulo m and n, respectively, we investigate conditions on A and B that ensure that, for at least some a in A and b in B, the linear system x = a mod m, x = b mod n has an integer solution, and we quantify the number of such admissible pairs (a,b). The special case where A and B consist of intervals of residue classes has application to the Lonely Runner Conjecture.
In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence $$ x^n\equiv λ\pmod p; \quad x\in \N, \quad L<x<L+p/n, $$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers $n, λ$ and $L$, for some absolute constant $c>0$. This implies, in part…
Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity, the proportion of the $r$-tuples of consecutive denominators with components in certain apriori fixed arithmetic progressions with the same ratio approaches a limit, which is independent on the interval. The limit is given explicitly and it is completely described in a few particular instances.
We prove several results which imply the following consequences. For any $\varepsilon>0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv 0\pmod p$, then the congruence $$ ax_1\cdots x_6+bx_7\cdots x_{13}\equiv c\pmod p $$ has a solution with $x_j\in\cI_j$. There exists an absolute constant $n_0\in\N$ such that for any $0<\varepsilon<1$ and any sufficiently large prime $p$, any quadratic residue $λ$ modulo $p$ can be represented in the form $$ x_1\cdots x_{n_0}\equiv λ\pmod p,\quad x_i\in\N,\quad x…
Given a primitive, non-CM, holomorphic cusp form $f$ with normalized Fourier coefficients $a(n)$ and given an interval $I\subset [-2, 2]$, we study the least prime $p$ such that $a(p)\in I$ . This can be viewed as a modular form analogue of Vinogradov's problem on the least quadratic non-residue. We obtain strong explicit bounds on $p$, depending on the analytic conductor of $f$ for some specific choices of $I$.
Using the Dirichlet theorem on the equidistribution of residue classes modulo $q$ and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes, we show that the domain of convergence of the infinite product of Dirichlet $L$-functions of non-principal characters can be extended from $\Re(s) > 1$ down to $\Re(s) > \half$, without encountering any zeros before reaching this critical line. The possibility of doing so can be traced back to a universal diffusive random walk behavior $C_N = {\cal O}(N^{1/2})$ of the series $C_N = \sum_{n=1}^N χ(p_n)$ ov…
