Estimates For Sums Research Articles

Average nonvanishing of Dirichlet L-functions at the central point

The Generalized Riemann Hypothesis implies that at least 50% of the central values $L \left( \frac{1}{2},\chi\right)$ are non-vanishing as $\chi$ ranges over primitive characters modulo $q$. We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo $q$ and averages $q$ over an interval, then at least 50.073% of the central values are non-vanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec.

Disjointness of the Möbius Transformation and Möbius Function

We study the distribution of the sequence of elements of the discrete dynamical system generated by the Mobius transformation $$x \mapsto (ax + b)/(cx + d)$$ over a finite field of p elements. Motivated by a recent conjecture of P. Sarnak, we obtain nontrivial estimates of exponential sums with such sequences that imply that trajectories of this dynamical system are disjoined with the Mobius function.

Distribution of $\alpha n +\beta $ modulo 1 over integers free from large and small primes

For any $\varepsilon \gt 0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert \alpha n + \beta \rVert \lt x^{-{1}/{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real numbers $x$. In addi

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.

Double exponential sums and congruences with intervals and exponential functions modulo a prime

Let p be a large prime number and g be any integer of multiplicative order T modulo p. We obtain a new estimate of the double exponential sumS=∑n∈N|∑m∈Mep(angm)|,gcd⁡(a,p)=1, where N and M are intervals of consecutive integers with |N|=N and |M|=M≤T elements. One representative example is the following consequence of the main result: if N=M≈p1/3, then |S|<N2−1/8+o(1). We then apply our estimate to obtain new results on additive congruences involving intervals and exponential functions.

On elementary estimates of arithmetic sums for polynomial rings over finite fields

In this paper, a simple and elementary method is given for deriving estimates of sums of arithmetic functions in Fq[t]. The method is the function field analogue of a result first proved by Stefan A. Burr in 1973 in the number field case. A novelty of this paper is that we are able to extend Burr's result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions dr(f) with an error term that improves the one given by Burr.

Comparison of some bounding inequalities applied in stability analysis of time-delay systems

This paper gives comparisons of several bounding inequalities proposed recently for the estimation of integrals and sums of quadratic functions, which plays an important role in stability analysis of time delay systems via the Lyapunov–Krasovskii approach. The equivalences of several known variants of the free matrix based inequalities and their generalized and simplified forms are shown. Secondly, the relationship between the (simplified) free matrix based inequality and the combination of the Bessel-based inequality with different bounding inequalities affine in the length of the intervals are investigated. Finally, we shall demonstrate the theoretical results with the application of the various estimations for the stability analysis of a linear system with time-varying delay using the same Lyapunov–Krasovskii functional.

Estimates for Character Sums in Finite Fields of Order p2 and p3

We obtain nontrivial bounds on character sums over “boxes” of volume pn(1/4+e) in finite fields of order pn for n = 2 and n = 3.

An Improved Remainder Estimate in the Weyl Formula for the Planar Disk

Y. Colin de Verdiere proved that the remainder term in the two-term Weyl formula for the eigenvalue counting function for the Dirichlet Laplacian associated with the planar disk is no more than $$O(\mu ^{2/3})$$ . In this paper, we study this problem from spectral geometry by using analysis and number theory. More precisely, by combining with the method of exponential sum estimation, we give a sharper remainder term estimate $$O(\mu ^{2/3-1/495})$$ .

Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

We study asymptotics of sums of the form ∑ X < n ⩽ 2 X Λ ( n ) Λ ( n + h ) , ∑ X < n ⩽ 2 X d k ( n ) d l ( n + h ) , ∑ X < n ⩽ 2 X Λ ( n ) d k ( n + h ) , and ∑ n Λ ( n ) Λ ( N − n ) , where Λ is the von Mangoldt function, d k is the kth divisor function, and N , X are large. Our main result is that the expected asymptotic for the first three sums holds for almost all h ∈ [ − H , H ] , provided that X σ + ε ⩽ H ⩽ X 1 − ε for some ε > 0 , where σ : = 8 33 = 0.2424 ⋯ , with an error term saving on average an arbitrary power of the logarithm over the trivial bound. This improves upon results of Mikawa and Baier–Browning–Marasingha–Zhao, who obtained statements of this form with σ replaced by 1 3 . We obtain an analogous result for the fourth sum for most N in an interval of the form [ X , X + H ] with X σ + ε ⩽ H ⩽ X 1 − ε . Our method starts with a variant of an argument from a paper of Zhan, using the circle method and some oscillatory integral estimates to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to ‘Type d 3 ’ and ‘Type d 4 ’ sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type d 3 sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type d 4 sum is treated similarly using the classical L 2 mean value theorem and the classical van der Corput exponential sum estimates. In a sequel to this paper we will obtain related results for the correlations involving d k ( n ) for much smaller values of H but with weaker bounds.

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.

On Double Sums with Multiplicative Characters

An estimate for double sums of multiplicative characters with additive convolutions of the characteristic functions of sets having small additive doubling is obtained.

Elementary Proof of an Estimate for Kloosterman Sums with Primes

A new elementary proof of an estimate for incomplete Kloosterman sums modulo a prime q is obtained. Along with Bourgain’s 2005 estimate of the double Kloosterman sum of a special form, it leads to an elementary derivation of an estimate for Kloosterman sums with primes for the case in which the length of the sum is of order q0.5+e, where e is an arbitrarily small fixed number.

Square-full primitive roots

We use character sum estimates to give some bounds on the least square-full primitive root modulo a prime. In particular, we show that there is a square-full primitive root mod [Formula: see text] less than [Formula: see text].

Mxpfit: A library for finding optimal multi-exponential approximations

mxpfit is a library implemented in C++ to find optimal approximations of functions by multi-exponential sums with complex-valued parameters. The library provides an interface for evaluating exponents and coefficients from sampling data on a uniform grid using the fast Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm originally proposed by Potts and Tasche (2015). The parameters can be estimated efficiently from a sampling data even including noise. A modified balanced truncation algorithm to find the multi-exponential sum with a smaller order is also provided. These features are useful for finding optimal exponential sum approximations of analytic functions or large-scale numerically sampled data set. Program summaryProgram Title: mxpfitProgram Files doi:http://dx.doi.org/10.17632/vjygkwpss4.1Licensing provisions: MITProgramming language: C++Nature of problem: Nonlinear curve fitting by multi-exponential function, frequency estimation of signals, and reduction of multi-exponential sumSolution method: The modified fast Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm originally proposed in [1] is utilized for the parameter estimation of multi-exponential sums from sampled data on a uniform grid. In this method, partial Lanczos bidiagonalization is used to find a low-rank approximation of a Hankel matrix constructed from sampled data. An overdetermined Vandermonde system is solved by the conjugate gradient least square (CGLS) method with preconditioner to obtain the coefficients of the multi-exponential sum. The truncation of the multi-exponential sum is performed by a modified balanced truncation method in [2]. The projection matrix to obtain the truncated system is determined by computing con-eigenvalues of a Gramian matrix, which can be computed efficiently with highly relative accuracy using the algorithm in [3].Additional comments: A C++ compiler supporting the C++11 standard is required to compile programs using the mxpfit library. mxpfit depends on the Eigen (http://eigen.tuxfamily.org/) and FFTW3 (http://www.fftw.org/) libraries. CMake (https://cmake.org/) is used for building example programs.

Poissonian pair correlation and discrepancy

A sequence (xn)n=1∞ on the torus T≅[0,1] is said to exhibit Poissonian pair correlation if the local gaps behave like the spacings of a Poisson random variable, i.e. limN→∞1N#1≤m≠n≤N:|xm−xn|≤sN=2salmost surely.We show that being close to Poissonian pair correlation for few values of s is enough to deduce global regularity statements: if, for some 0<δ<1∕2, a set of points x1,…,xN satisfies 1N#1≤m≠n≤N:|xm−xn|≤sN≤(1+δ)2sfor all1≤s≤(8∕δ)logN,then the discrepancy DN of the set satisfies DN≲δ1∕3+N−1∕3δ−1∕2. We also show that distribution properties are reflected in the global deviation from the Poissonian pair correlation N2DN5≲2N∫0N∕21N#1≤m≠n≤N:|xm−xn|≤sN−2s2ds≲N2DN2,where the lower bound is conditioned on DN≳N−1∕3. The proofs use a connection between exponential sums, the heat kernel on T and spatial localization. Exponential sum estimates are obtained as a byproduct. We also describe a connection to diaphony and several open problems.

Estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis containing the Möbius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the Möbius function. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.

Elementary Proof of an Estimate for Kloosterman Sums with Primes

В работе получено новое элементарное доказательство оценки неполной суммы Клоостермана по простому модулю $q$. Наряду с полученной Ж. Бургейном (2005) оценкой двойной суммы Клоостермана специального вида, оно приводит к элементарному выводу оценки суммы Клоостермана с простыми числами в случае, когда длина суммы имеет порядок $q^{0.5+\varepsilon}$, где $\varepsilon$ - сколь угодно малое фиксированное число. Библиография: 18 названий.

An exponential sum estimate for systems with linear polynomials

An exponential sum estimate for systems with linear polynomials

СУММЫ ХАРАКТЕРОВ НА СДВИНУТЫХ СТЕПЕНЯХ

We study character sums over shifted powers modulo a prime p. Such sums can be viewed as generalizations of character sums over shifted multiplicative subgroups. We obtain some new results on upper estimates for absolute value of these sums. The case when the cardinality of subgroup is less than √p, it is a question of non-trivial upper bounds for such sums that remains open and is unsolved today. It was proposed by J. Burgain and M. Ch. Chang in the review of 2010. Nevertheless, some intermediate results were achieved by Professor K. Gong, who established non-trivial estimates of such sums in the case when the subgroup is much larger than √p. In this paper, we obtain some new results on the upper bound for the absolute value of the generalization of such sums, which are incomplete sums of character sums over shifted subgroups. Two proofs of the main result are given. The first one is based on reduction of this sum to the well-known estimate of A. Weil and the method of smoothing such sums. The method of estimating the incomplete sum through the full one is also applied. One result of M. Z. Garaev is also used. The second proof is based on the original idea of I. M. Vinogradov. This approach was proposed to refine the known inequality of Poya-Vinogradov and uses in its essence some geometric and combinatorial ideas. The second proof is not fully presented. We only prove a key statement, and for the rest of the calculations we refer the reader to the initial work of I. M. Vinogradov.