Short Sums Research Articles

A Pólya–Vinogradov inequality for short character sums

AbstractIn this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume $\chi $ to be a primitive character modulo q, $ \epsilon>0$ and $N\le q^{1-\gamma }$ , with $0\le \gamma \le 1/3$ . We prove that $$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$ with $c=2/\pi ^2$ if $\chi $ is even and $c=1/\pi $ if $\chi $ is odd. The result is based on the work of Hildebrand and Kerr.

Convergence acceleration of Monte Carlo many-body perturbation methods by direct sampling.

In the Monte Carlo many-body perturbation (MC-MP) method, the conventional correlation-correction formula, which is a long sum of products of low-dimensional integrals, is first recast into a short sum of high-dimensional integrals over electron-pair and imaginary-time coordinates. These high-dimensional integrals are then evaluated by the Monte Carlo method with random coordinates generated by the Metropolis-Hasting algorithm according to a suitable distribution. The latter algorithm, while advantageous in its ability to sample nearly any distribution, introduces autocorrelation in sampled coordinates, which, in turn, increases the statistical uncertainty of the integrals and thus the computational cost. It also involves wasteful rejected moves and an initial "burn-in" step as well as displays hysteresis. Here, an algorithm is proposed that directly produces a random sequence of electron-pair coordinates for the same distribution used in the MC-MP method, which is free from autocorrelation, rejected moves, a burn-in step, or hysteresis. This direct-sampling algorithm is shown to accelerate second- and third-order Monte Carlo many-body perturbation calculations by up to 222% and 38%, respectively.

Burgess bounds for short character sums evaluated at forms

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 $q^{\beta}$, with $\beta > 1/2 - 1/(2(n+1))$. 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 second author.

On fourth and higher moments of short exponential sums related to cusp forms

We obtain upper bounds for the fourth and higher moments of short exponential sums involving Fourier coefficients of holomorphic cusp forms twisted by rational additive twists with small denominators. We obtain the conjectured best possible bound in the case of the fourth moment.

The maximum size of short character sums

In the present note, we prove new lower bounds on large values of character sums $$\varDelta (x,q):=\max _{\chi \ne \chi _0} \big \vert \sum _{n\le x} \chi (n)\big \vert $$ in certain ranges of x. Employing an implementation of the resonance method developed in a work involving the author in order to exhibit large values of L-functions, we improve some results of Hough in the range $$\log x = o\big (\sqrt{\log q}\big )$$ . Our results are expressed using the counting function of y-friable integers less than x, where we improve the level of smoothness y for short intervals.

The 16-rank of ℚ()

Recently, a density result for the $16$-rank of $\text{Cl}(\mathbb{Q}(\sqrt{-p}))$ was established when $p$ varies among the prime numbers, assuming a short character sum conjecture. In this paper we prove the same density result unconditionally.

A conjecture of Erdős, supersingular primes and short character sums

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation \[ n(n+d)\cdots(n+(k-1)d) = y^\ell \] has at most finitely many solutions in positive integers $n$, $d$, $y$ and $\ell$, with $\mathrm{gcd}(n,d)=1$ and $\ell \ge 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.

Sums of algebraic trace functions twisted by arithmetic functions

We obtain new bounds for short sums of isotypic trace functions associated to some sheaf modulo prime $p$ of bounded conductor, twisted by the Mobius function and also by the generalised divisor function. These trace functions include Kloosterman sums and several other classical number theoretic objects. Our bounds are nontrivial for intervals of length at least $p^{1/2+\varepsilon}$ with an arbitrary fixed $\varepsilon >0$, which is shorter than the length at least $p^{3/4+\varepsilon}$ in the case of the Mobius function and at least $p^{2/3+\varepsilon}$ in the case of the divisor function required in recent results of {\'E}.~Fouvry, E.~Kowalski and P.~Michel (2014) and E.~Kowalski, P. ~Michel and W.~Sawin (2018), respectively.

Distribution of short sums of classical Kloosterman sums of prime powers moduli

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are P.~Deligne's equidistribution theorem, N.~Katz' works and the results surveyed in \cite{MR3338119}. In particular, this applies to $2$-dimensional Kloosterman sums $\mathsf{Kl}_{2,\mathbb{F}_q}$ studied by N.~Katz in \cite{MR955052} and in \cite{MR1081536} when the field $\mathbb{F}_q$ gets large. \par This article considers the case of short sums of normalized classical Kloosterman sums of prime powers moduli $\mathsf{Kl}_{p^n}$, as $p$ tends to infinity among the prime numbers and $n\geq 2$ is a fixed integer. A convergence in law towards a real-valued standard Gaussian random variable is proved under some very natural conditions.

Bounds on short character sums and L-functions with characters to a powerful modulus

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 Πp|qp. 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 classical zero-free region (which might include Siegel zeros). Finally, we improve the previous best value $$L=\frac{12}{5}=2.2$$ of the Linnik constant for primes in arithmetic progressions modulo powers of a fixed prime to L < 2.1115.

Sums with the Möbius function twisted by characters with powerful moduli

In a 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 with the Möbius function twisted by characters of the same type, which complements and improves 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 L -functions with the same class of moduli.

Distributed Processing of Location-Based Aggregate Queries Using MapReduce

The location-based aggregate queries, consisting of the shortest average distance query (SAvgDQ), the shortest minimal distance query (SMinDQ), the shortest maximal distance query (SMaxDQ), and the shortest sum distance query (SSumDQ) are new types of location-based queries. Such queries can be used to provide the user with useful object information by considering both the spatial closeness of objects to the query object and the neighboring relationship between objects. Due to a large amount of location-based aggregate queries that need to be evaluated concurrently, the centralized processing system would suffer a heavy query load, leading eventually to poor performance. As a result, in this paper, we focus on developing the distributed processing technique to answer multiple location-based aggregate queries, based on the MapReduce platform. We first design a grid structure to manage information of objects by taking into account the storage balance, and then develop a distributed processing algorithm, namely the MapReduce-based aggregate query algorithm (MRAggQ algorithm), to efficiently process the location-based aggregate queries in a distributed manner. Extensive experiments using synthetic and real datasets are conducted to demonstrate the scalability and the efficiency of the proposed processing algorithm.

Multifactorial intervention for hip and pelvic fracture patients with mild to moderate cognitive impairment: study protocol of a dual-centre randomised controlled trial (OF-CARE)

BackgroundA hip or pelvic fracture is a major fall-related injury which often causes a decline in mobility performance and physical activity. Over 40% of patients with hip fracture have cognitive impairment or dementia and poorer rehabilitation outcomes than those without cognitive impairment. In this subgroup, there is a lack of evidence on the best practices supporting recovery. The main aim of this study is to investigate the effects of a transitional care intervention after inpatient rehabilitation on physical activity and functional performance in this group of cognitively impaired patients.Methods/designThis dual-centre, randomised controlled trial compares a multifactorial intervention with usual care as control condition. Two hundred and forty community-dwellers (≥ 65 years) with a hip or pelvic fracture and mild to moderate cognitive impairment (MMSE 17–26) are recruited at the end of inpatient rehabilitation.The four-month intervention consists of (a) an individually tailored, progressive home exercise program and physical activity promotion delivered by professional instructors and lay instructors (two home visits per week) and (b) a long-term care counselling approach addressing unmet care needs, pleasurable activities, and caregiver issues if needed. Primary outcome parameters are physical activity, measured as daily walking duration with an accelerometer-based activity monitor (activPAL™) over 72 h, and functional performance, assessed with Short Physical Performance Battery sum scores. Secondary outcome parameters are fear of falling, fall related self-efficacy, falls, quality of life, depression and activity of daily living. Data are collected at the end of rehabilitation, before the intervention at the patient’s home (baseline), after four months (post-intervention), and seven months (follow-up). In addition to completer and intent-to-treat analyses of outcomes, economic data and incremental cost-effectiveness are analysed.DiscussionExisting service models of volunteer services and legal counselling provided by care counsellors were considered when developing the intervention protocol. Therefore, it should be feasible to translate and deliver the intervention into real-world practice if it has been demonstrated to be effective.Trial registrationGerman Clinical Trials Register, DRKS00008863 (Accessed 17 Apr 2019), ISRCTN registry, ISRCTN69957256 (Accessed 17 Apr 2019).

Diophantine approximation on the parabola with non-monotonic approximation functions

AbstractWe show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarník type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess’s bound on short character sums.

Exponential sums related to Maass forms

We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms. The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously proven for holomorphic cusp form coefficients. As an application of these, we remove the logarithm from the classical upper bound for long linear sums weighted by Fourier coefficients of Maass forms, the resulting estimate being the best possible. This also involves improving the upper bounds for long linear sums with rational additive twists, the gains again allowed by the estimates for the short sums. Finally, we shall use the approximate functional equation to bound somewhat longer short exponential sums.

About the life and activities of academician Zarullo Husenovich Rakhmonov

The article is devoted to the doctor of physico-mathematical Sciences, academician of the Academy of Sciences of the Republic of Tajikistan, one of the foremost experts in the field of number theory, Zarullo Husenovich Rakhmonov in connection with his 60–year anniversary. Provides a brief biography, the main stages of development of his scientific career. We give the review of results of Z. H. Rakhmonov on following problems: on the distribution of the Goldbach’s and Hardy–Littlewood’s numbers in short arithmetical progressions, on the problem of mean values of the Chebyshev’s function and the problem of the Riemann zeta-function zeros belonging to short rectangular in the critical strip, to estimations short trigonometric sums over primes and on the Goldbach’s problem with almost equals summands, on the Selberg’s problem concerning to the Riemann’s zeta-function zeros lying on short intevals of the critical line. In conclusion, the author presents a list of main scientific publications Z. H. Rakhmonov

Spins of prime ideals and the negative Pell equation

Let$p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4$be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group$\text{Cl}(-4p)$of the imaginary quadratic number field$\mathbb{Q}(\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group$\text{Cl}(8p)$of the real quadratic field$\mathbb{Q}(\sqrt{8p})$; (iii) the solvability of the negative Pell equation$x^{2}-2py^{2}=-1$over the integers; (iv) $2$-part of the Tate–Šafarevič group$\unicode[STIX]{x0428}(E_{p})$of the congruent number elliptic curve$E_{p}:y^{2}=x^{3}-p^{2}x$. Our results are conditional on a standard conjecture about short character sums.

Large values of short character sums

TextIn this paper, we prove that for any A>0 there exist infinitely many primes p for which sums of the Legendre symbols modulo p over an interval of length (ln⁡p)A can take large values. VideoFor a video summary of this paper, please visit https://youtu.be/wNvv8wujF3o.

Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums

In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the sequence of classical Fourier coefficients by a short exponential sum with a pre-defined number of $$N+1$$ terms. The obtained approximation can be interpreted as an adaptive N-th Fourier sum with respect to the orthogonal Takenaka-Malmquist basis. Using the theoretical results on rational approximation in Hardy spaces and on the decay of singular values of special infinite Hankel matrices, we show that adaptive Fourier sums can converge essentially faster than classical Fourier sums for a large class of functions. Further, we derive an algorithm to compute almost optimal adaptive Fourier sums. Our numerical results show that the significantly better convergence behavior of adaptive Fourier sums for optimally chosen basis elements can also be achieved in practice. For comparison, we also provide a greedy algorithm to determine an adaptive Fourier sum. This algorithm requires less computational effort but yields essentially slower convergence.

Distribution questions for trace functions with values in cyclotomic integers and their reductions

We consider ℓ \ell -adic trace functions over finite fields taking values in cyclotomic integers, such as characters and exponential sums. Through ideas of Deligne and Katz, we explore probabilistic properties of the reductions modulo a prime ideal, exploiting especially the determination of their integral monodromy groups. In particular, this gives a generalization of a result of Lamzouri-Zaharescu on the distribution of short sums of the Legendre symbol reduced modulo an integer to all multiplicative characters and to hyper-Kloosterman sums.