Asymptotic formula
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Approximation properties of λ-Bernstein operators Open
In this paper, we introduce a new type λ-Bernstein operators with parameter [Formula: see text], we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz co…
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Moments of zeta and correlations of divisor-sums: III Open
In this series we examine the calculation of the 2kth moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper is concerned with the precise i…
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Some Approximation Results on $\lambda-$ Szasz-Mirakjan-Kantorovich Operators Open
In this article, we purpose to obtain several approximation properties of Sz\'{a}sz-Mirakjan-Kantorovich operators with shape parameter $\lambda \in\lbrack-1,1]$. We compute some preliminaries results such as moments and central moments fo…
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Squares: Additive questions and partitions Open
This memoir is concerned with a number of additive questions in which squares occur. For ternary additive problems with at least one square, the exceptional set is considered and it is shown that current methods cannot be sharpened substan…
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Asymptotic resurgence via integral closures Open
Given an ideal in a polynomial ring, we show that the asymptotic resurgence studied by Guardo, Harbourne, and Van Tuyl can be computed using integral closures. As a consequence, the asymptotic resurgence of an ideal is the maximum of finit…
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Integer partitions, probabilities and quantum modular forms Open
What is the probability that the smallest part of a random integer partition of N is odd? What is the expected value of the smallest part of a random integer partition of N? It is straightforward to see that the answers to these questions …
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Multigrade efficient congruencing and Vinogradov's mean value theorem Open
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 $\tfrac{5}{8}k^21.543k^2$. The asymptotic form…
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Linear Wave Interaction with a Vertical Cylinder of Arbitrary Cross Section: An Asymptotic Approach Open
An asymptotic approach to the linear problem of regular water waves interacting with a vertical cylinder of an arbitrary cross section is presented. The incident regular wave was one-dimensional, water was of finite depth, and the rigid cy…
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Mean value theorems for the double zeta-function Open
We prove asymptotic formulas for mean square values of the Euler double zeta-function $\\zeta_2(s_0,s)$, with respect to $\\Im s$. Those formulas enable us to propose a double analogue of the Lindelöf hypothesis.
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Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain Open
We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reprodu…
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Linear correlations of multiplicative functions Open
We prove a Green–Tao type theorem for multiplicative functions. Asymptotic results on expressions of the form 1.1 are known in many special cases: Green and Tao 14 establish such results for the Möbius function μ. Using the machinery from …
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Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$ Open
In this paper, we study several approximation properties of Szasz-Mirakjan-Durrmeyer operators with shape parameter λ∈[−1,1]λ∈[−1,1]. Firstly, we obtain some preliminaries results such as moments and central moments. Next, we estimate the …
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( p , q ) $(p,q)$ -gamma operators which preserve x 2 $x^{2}$ Open
In this paper, we introduce (p,q) $(p,q)$-gamma operators which preserve x2 $x^{2}$, we estimate the moments of these operators, and establish direct and local approximation theorems of these operators. Then two approximation theorems abou…
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Power‐free values of polynomials Open
Let be an irreducible polynomial of degree with no fixed th power prime divisor. We derive an asymptotic formula for the number of primes such that is ‐free.
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The variance of divisor sums in arithmetic progressions Open
We study the variance of sums of the k -fold divisor function d k ( n ) {d_{k}(n)} over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of…
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A new fourth power mean of two-term exponential sums Open
The main purpose of this paper is to use analytic methods and properties of quartic Gauss sums to study a special fourth power mean of a two-term exponential sums mod p , with p an odd prime, and prove interesting new identities. As an app…
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Distribution of orders in number fields Open
In this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our study, we use p-adic and motivic integration techn…
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On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity Open
The paper is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second-order elliptic p…
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Modular invariants for real quadratic fields and Kloosterman sums Open
We investigate the asymptotic distribution of integrals of the $j$-function\nthat are associated to ideal classes in a real quadratic field. To estimate the\nerror term in our asymptotic formula, we prove a bound for sums of Kloosterman\ns…
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High moments of the Estermann function Open
For $a/q\\in\\mathbb{Q}$ the Estermann function is defined as\n$D(s,a/q):=\\sum_{n\\geq1}d(n)n^{-s}\\operatorname{e}(n\\frac aq)$ if $\\Re(s)>1$\nand by meromorphic continuation otherwise. For $q$ prime, we compute the\nmoments of $D(s,a/q…
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The second moment theory of families of L-functions Open
For a fairly general family of L-functions, we survey the known consequences of the existence of asymptotic formulas with power-sawing error term for the (twisted) first and second moments of the central values in the family. We then consi…
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Mathematical Problems in Quantum Physics Open
International audience
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Durrmeyer-type generalization of μ-Bernstein operators Open
In the present manuscript, we consider ?-Bernstein-Durrmeyer operators involving a strictly positive continuous function. Firstly, we prove a Voronovskaja type, quantitative Voronovskaja type and Gr?ss-Voronovskaja type asymptotic formula,…
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The mean values of cubic L-functions overfunction fields Open
We obtain an asymptotic formula for the mean value of L-functions associated\nto cubic characters over F_q[t]. We solve this problem in the non-Kummer\nsetting when q=2 (mod 3) and in the Kummer case when q=1 (mod 3). The proofs\nrely on o…
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The Euler binary partition function and subdivision schemes Open
For an arbitrary set $D$ of nonnegative integers, we consider the Euler binary partition function $b(k)$ which equals the total number of binary expansions of an integer $k$ with "digits" from $D$. By applying the theory of subdivision sch…
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Slicing the stars: counting algebraic numbers, integers, and units by degree and height Open
Masser and Vaaler have given an asymptotic formula for the number of\nalgebraic numbers of given degree $d$ and increasing height. This problem was\nsolved by counting lattice points (which correspond to minimal polynomials over\n$\\mathbb…
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Approximation by generalized Stancu type integral operators involving Sheffer polynomials Open
In this article, we give a generalization of integral operators which involves Sheffer polynomials introduced by Sucu and Buy¨ ukyazici. We obtain approximation properties of our operators with the help of the univer- ¨ sal Korovkin’s theo…
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ON MINIMAL ASYMPTOTIC -ADIC BASES Open
Let $g\geq 2$ be a fixed integer. Let $\mathbb{N}$ denote the set of all nonnegative integers and let $A$ be a subset of $\mathbb{N}$ . Write $r_{2}(A,n)=\sharp \{(a_{1},a_{2})\in A^{2}:a_{1}+a_{2}=n\}.$ We construct a thin, strongly minim…
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Combinatorial identities and Titchmarsh’sdivisor problem for multiplicative functions Open
Given a multiplicative function $f$ which is periodic over the primes, we\nobtain a full asymptotic expansion for the shifted convolution sum\n$\\sum_{|h|<n\\leq x} f(n) \\tau(n-h)$, where $\\tau$ denotes the divisor function\nand $h\\in\\…
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Convergence estimates of a family of approximation operators of exponential type Open
The main object of this article is to consider a family of approximation operators of exponential type, which has presumably not been studied earlier due mainly to their seemingly complicated behavior. We estimate and establish a quantitat…