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 an explicit Burgess-like result on convoluted Dirichlet characters.
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the GRH. We give a simple proof of their estimate, and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large,…
We give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov's mean value. Coupling this with the recent works of Wooley, and of Bourgain, Demeter and Guth, providing optimal bounds for the Vinogradov mean value, we produce a powerful new $k$-th derivative estimate. Roughly speaking, this improves the van der Corput estimate for $k\ge 4$. Various corollaries are given, showing for example that $ζ(σ+it)\ll_{\varepsilon}t^{(1-σ)^{3/2}/2+\varepsilon}$ for $t\ge 2$ and $0\leσ\le 1$, for any fixed $\varepsilon>0$.
In this paper we obtain a variation of the Pólya--Vinogradov inequality with the sum restricted to a certain height. Assume $χ$ to be a primitive character modulo $q$, $ε> 0$ and $N\le q^{1-γ}$, with $0\le γ\le 1/3$. We prove that \begin{equation*} \left|\sum_{n=1}^N χ(n) \right|\le c(\frac{1}{3}-γ+ε)\sqrt{q}\log q \end{equation*} with $c=2/π^2+o(1)$ if $χ$ is even and $c=1/π+o(1)$ if $χ$ is odd.
We show in a quantitative way that any odd character $χ$ modulo $q$ of fixed order $g \geq 2$ satisfies the property that if the Pólya-Vinogradov inequality for $χ$ can be improved to $$\max_{1 \leq t \leq q} \left|\sum_{n \leq t} χ(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q)$$ then for any $ε> 0$ one may exhibit cancellation in partial sums of $χ$ on the interval $[1,t]$ whenever $t > q^ε$, i.e.,$$\sum_{n \leq t} χ(n) = o_{q \rightarrow \infty}(t) \text{ for all $t > q^ε$.}$$ This generalizes and extends a result of Fromm and Goldmakher. We also prove a converse implication, to the e…
In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for squarefree modulus. Given a primitive character $χ$ to squarefree modulus $q$, 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 $c=1/(2π^2)+o(1)$ for even characters and $c=1/(4π)+o(1)$ for odd characters, with an explicit $o(1)$ term. This improves a result of Frolenkov and Soundararajan for large $q$. We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with mul…
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.
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.
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…
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.
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.
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 $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.
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.
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.
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…
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.
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 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.
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.
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 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.
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.
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.
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.
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.
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.
In this paper, we investigate the distribution of the maximum of partial sums of families of $m$-periodic complex valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$-adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are…
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 …
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…
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…
For "almost all" sufficiently large $N,$ satisfying necessary congruence conditions and $k\geq 2$, we show that there is an {\bf asymptotic formula} for the number of solutions of the equation \begin{align*} \begin{split} &N=p_{1}^{k}+p_{2}^{k}+\cdots+p_{s}^{k}, \\ &\left|p_{i}-( N/s)^{1/k}\right|\leq (N/s)^{θ/k},\ (1\leq i\leq s) \end{split} \end{align*} with \begin{align*} s\geq \frac{k(k+1)}{2}+1\ \ \textup{and}\ \ θ\geq {\bf 2/3}+\varepsilon. \end{align*} This enlarges the effective range of $s$ for which can be obtained by the method of Mätomaki and Xuancheng Shao \cite{MS}. [Discorrelati…
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…
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.
