Classical Sum Research Articles

On some explicit evaluations of nonlinear Euler sums

In this paper, we extend tools developed in [26] to study harmonic numbers sums. We use these objects to compute Euler-related sums and to explore their connections with classical Euler sums. We establish some new relations between Euler sums and related sums. The relations obtained allow us to find some nice closed-form representations of nonlinear Euler sums through zeta values and linear sums.

Evaluations of Euler-Type Sums of Weight $$\le 5$$≤5

Let $$p,p_1,\ldots ,p_m$$ be positive integers with $$p_1\le p_2\le \cdots \le p_m$$ and $$x\in [-1,1)$$, define the so-called Euler-type sums $${S_{{p_1}{p_2} \ldots {p_m},p}}\left( x \right) $$, which are the infinite sums whose general term is a product of harmonic numbers of index n, a power of $$n^{-1}$$ and variable $$x^n$$, by $$\begin{aligned} {S_{{p_1}{p_2} \ldots {p_m},p}}\left( x \right) : = \sum \limits _{n = 1}^{\infty } {\frac{{H_n^{\left( {{p_1}} \right) }H_n^{\left( {{p_2}} \right) } \ldots H_n^{\left( {{p_m}} \right) }}}{{{n^p}}}{x^n}}\quad (m\in \mathbb {N}:=\{1,2,3,\ldots \}), \end{aligned}$$where $$H_n^{(p)}$$ is defined by the generalized harmonic number. Extending earlier work about classical Euler sums, we prove that whenever $$p+p_1+\cdots +p_m \le 5$$, then all sums $${S_{{p_1}{p_2} \ldots {p_m},p}}\left( 1/2\right) $$ can be expressed as a rational linear combination of products of zeta values, polylogarithms and $$\log (2)$$. The proof involves finding and solving linear equations which relate the different types of sums to each other.

New Data on Photoneutron Reaction Cross Sections for 76,78,80,82Se Nuclei

The problem of reliability of the cross section data obtained for partial photoneutron reactions on 76,78,80,82Se nuclei in beams of quasimonoenergetic annihilation photons by means of neutron multiplicity sorting is discussed by employing objective physical criteria. It is shown that, because of substantial systematic uncertainties, experimental data on the (γ, 1n) and (γ, 2n) cross sections are unreliable. New data satisfying the reliability criteria are obtained for the partial photoneutron reaction cross sections for 76,78,82Se nuclei by an experimental—theoretical method for evaluating such cross sections and are compared with experimental data and with data evaluated earlier for the isotope 80Se. The evaluated integrated cross sections for the total photoneutron reactions on 76,78,80,82Se nuclei are compared with the predictions of the Thomas-Reiche-Kuhn classical dipole sum rule.

О некоторых обобщениях суммы степеней натуральных чисел

In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The so-called binomial sums are also considered. The problem of constructing polynomials that allow to calculate the values of the corresponding sums in certain cases is solved.

A Note on the Classical Gauss Sums

The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums mod p ( a prime with p ≡ 1 mod 12 ) to give an exact evaluation formula for it.

On Classical Gauss Sums and Some of Their Properties

The goal of this paper is to solve the computational problem of one kind rational polynomials of classical Gauss sums, applying the analytic means and the properties of the character sums. Finally, we will calculate a meaningful recursive formula for it.

On the classical Gauss sum and the recursive properties

Let p be a prime with pequiv1bmod8, ψ be an eighth character modp, and tau(psi) denote the classical Gauss sum modp. The main purpose of this paper is using the analytic method and the properties of the classical Gauss sum to study the computational problem of one kind rational polynomial of tau(psi). In the end, we prove an interesting second-order linear recursive formula for it.

Tiling, Circle Packing and Exponential Sums over Finite Fields

We study the problem of tiling and packing in vector spaces over finite fields and its connections with zeroes of classical exponential sums. In particular, we study tilings mostly in two and three dimensions and packings in dimension two. A combination of Fourier analytic and algebraic methods is employed.

On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums

Abstract The main purpose of this paper is to study the computational problem of one kind hybrid power mean involving two-term exponential sums and quartic Gauss sums using the analytic method and the properties of the classical Gauss sums, and to prove some interesting fourth-order linear recurrence formulae for this problem. As an application of our result, we can also obtain an exact computational formula for one kind congruence equation mod p, an odd prime.

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.

The distribution of consecutive prime biases and sums of sawtooth random variables

AbstractIn recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

Characterization of Kollár surfaces

Koll\'ar introduced in [Ko08] the surfaces $$(x_1^{a_1}x_2+x_2^{a_2}x_3+x_3^{a_3}x_4+x_4^{a_4}x_1=0)\subset \mathbb{P}(w_1,w_2,w_3,w_4)$$ where $w_i=W_i/w^*$, $W_i=a_{i+1}a_{i+2}a_{i+3}-a_{i+2}a_{i+3}+a_{i+3}-1$, and $w^*=$gcd$(W_1,\ldots,W_4)$. The aim was to give many interesting examples of $\mathbb{Q}$-homology projective planes. They occur when $w^*=1$. For that case, we prove that Koll\'ar surfaces are Hwang-Keum [HK12] surfaces. For $w^*>1$, we construct a geometrically explicit birational map between Koll\'ar surfaces and cyclic covers $z^{w^*}=l_1^{a_2 a_3 a_4} l_2^{-a_3 a_4} l_3^{a_4} l_4^{-1}$, where $\{l_1,l_2,l_3,l_4\}$ are four general lines in $\mathbb{P}^2$. In addition, by using various properties on classical Dedekind sums, we prove that: (a) For any $w^*>1$, we have $p_g=0$ iff the Koll\'ar surface is rational. This happens when $a_{i+1} \equiv 1$ or $a_{i}a_{i+1} \equiv -1 ($mod $w^*)$ for some $i$. (b) For any $w^*>1$, we have $p_g=1$ iff the Koll\'ar surface is birational to a K3 surface. We classify this situation. (c) For $w^*>>0$, we have that the smooth minimal model $S$ of a generic Koll\'ar surface is of general type with $K_{S}^2/e(S) \to 1$.

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.

Dedekind sums take each value infinitely many times

For [Formula: see text] and [Formula: see text], [Formula: see text], let [Formula: see text] denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs [Formula: see text], [Formula: see text], with [Formula: see text] tending to infinity as [Formula: see text] grows, such that [Formula: see text] for all [Formula: see text].

The switch operators and push-the-button games: A sequential compound over rulesets

We study operators that combine combinatorial games. This field was initiated by Sprague–Grundy (1930s), Milnor (1950s) and Berlekamp–Conway–Guy (1970–80s) via the now classical disjunctive sum operator on (abstract) games. The new class consists in operators for rulesets, dubbed the switch-operators. The ordered pair of rulesets (R1,R2) is compatible if, given any position in R1, there is a description of how to move in R2. Given compatible (R1,R2), we build the push-the-button game R1⊚R2, where players start by playing according to the rules R1, but at some point during play, one of the players must switch the rules to R2, by pushing the button ‘⊚’. Thus, the game ends according to the terminal condition of ruleset R2. We study the pairwise combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we prove that standard periodicity results for Subtraction games transfer to this setting, and we give partial results for a variation of Domineering, where R1 is the game where the players put the domino tiles horizontally and R2 the game where they play vertically (thus generalizing the octal game 0.07).

Accumulated Angle Factor-Based Beamforming to Improve the Visualization of Spinal Structures in Ultrasound Images.

In recent years, ultrasound has been increasingly used to guide needle insertion procedures for spinal anesthesia. The primary anatomical targets are facet joints and epidural spaces. For these procedures, accurate visualization of the spine anatomy is of critical importance. Challenges arising from the interactions between the ultrasound beam and spinal structures including tilt caused by specular reflections, off-axis interference, and reverberations often result in weakened and blurred vertebra surfaces. Previously, adaptive beamforming methods have been proposed to improve the resolution and contrast. However, most of these methods are not specialized for improving the contrast of specular targets like bones. In this paper, we propose an accumulated angle factor (AAF)-based beamforming method customized for bone surface enhancement. This approach applies a Hilbert transform on delay compensated channel data across the receive aperture. The accumulated phase change across the receive aperture is then calculated and utilized as the weight in the beamforming output. We compared our method with classical delay and sum (DAS) beamforming method and adaptive beamforming methods such as Wiener, phase coherence factor (PCF), CF, and generalized CF (GCF) beamforming. In 12 volunteer data sets, the mean contrast ratio between the vertebrae surface and the surrounding tissue for DAS, Wiener, PCF, CF, GCF, and the proposed AAF methods are 0.49, 0.64, 0.82, 0.77, 0.76, and 0.91, respectively. The contrast is significantly improved in the proposed method.

Equivariant Gauss sum of finite quadratic forms

Abstract The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulas hold for some basic classes of quadratic forms. In applications, such invariant appears in the dimension formula for certain vector-valued modular forms.

Ramanujan expansions of arithmetic functions of several variables

We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel.

On the fourth-order linear recurrence formula related to classical Gauss sums

Abstract Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ be any fourth-order character mod p. In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G(k, p) = τk(ψ)+τk(ψ), and give an interesting fourth-order linear recurrence formula for it, where τ(ψ) denotes the classical Gauss sums.

An alternative procedure to cumulative sum for cyber‐physical attack detection

Motivated by the importance of change detection in cyber‐physical systems security, an alternative procedure to the classical cumulative sum (CUSUM) algorithm is explored. By capitalizing on recent results appeared in the statistical literature, it is shown that superior performance can be obtained in terms of reaction time while being robust against parameter mismatches. In order to translate such quickest‐detection‐theoretic advances into practical system‐engineering tools, a simple pseudocode is provided that ensures approximately flat response with respect to the change time. We believe the latter is a very important property that is missing in CUSUM‐based detection approaches.