VINOGRADOV'S INTEGRAL AND BOUNDS FOR THE RIEMANN ZETA FUNCTION

In this paper, we prove new ℓp→ℓq bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number rs, k(l) of representations of a positive integer l as a sum of s positive kth powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a further range of complementary results for the discrete operator under consideration.

Highly parallel computations modulo a number having only small prime factors

We give upper bounds for the absolute value of exponential sums in several variables attached to certain polynomials with coefficients in a finite field. This bounds are given in terms of invariants of the singularities of the projective hypersurface defined by its highest degree form. For exponential sums attached to the reduction modulo a power of a large prime of a polynomial f with integer coefficients and veryfying a certain condition on the singularities of its highest degree form, we give a bound in terms of the dimension of the Jacobian quotient .

The article is devoted to the problem of the real numbers' decomposition to a linear combination of natural numbers' squares. First of all the theorem giving indirectly the conditions of existing natural numbers' two equal squares is formulated and proved. Basing on this theorem two algorithms designed for finding couples of integer numbers generating eąual sums of sąuares and determining the multiplicity of that couples are presented. Finally the calculation of the algorithms' classes and estimation of the calculation errors is made. Also an example output of presented algorithms was shown.

Si q≥2 est un nombre entier, on designe par Sq(n) la somme des chiffres en base q du nombre entier naturel n et par vq(n) sa valuation q-adique. L’objectif de cet article est d’etudier des sommes d’exponentielles de la forme ∑n≤xexp(2iπ(lmSq(n)+km′Sq(n+1)+θn)) afin d’en deduire certaines proprietes statistiques des nombres entiers n pour lesquels Sq(n) et Sq(n+1) appartiennent a des progressions arithmetiques donnees. Ceci permet d’etendre les resultats obtenus par Gelfond en 1968 et ceux obtenus par Mauduit-Sarkozy en 1996.

Suppose {X(t), t ∈ G d } is a random field in d dimensions, with d ∈ ℤ+; that is, {X(t), t ∈ G d } is a collection of random variables X(t) taking values in a state space S, defined on a probability space (Ω, A, P), and indexed by the variable t ∈ G d . Throughout this chapter, G will stand for either the set of real numbers ℝ, or the set of integers ℤ; thus, the random field {X(t)} is allowed to “run” in either continuous or discrete “time.” Similarly, G+ will denote ℝ+ or ℤ+ (the sets of positive real numbers and integers, respectively) according to whether G = ℝ or G = ℤ.

Mean value of some exponential sums and applications to Kloosterman sums

Real or natural number interpretation and their effect on complexity

In our previous work (Y. Hashimoto, Selberg’s zeta function for the modular group in the critical strip, Math. Nachr. 294 (2021) 1899–1904, https://doi.org/10.1002/mana.202000268 ), we proposed an upper bound of the logarithmic derivative of Selberg’s zeta function for the modular group in the critical strip. This paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups.

Several classes of exponential sums and three-valued Walsh spectrums over finite fields

n= Zx,+ i=1 The principal tool in the proof is the Hardy-Littlewood circle method, incorporating results and techniques of a powerful new iterative method developed by Vaughan and Wooley ([Va3], [Va4], [VW1], [Wol], [VW2]). In section 2, an algorithm developed in [Fo] for optimizing the parameters in mixed power mean value theorems is generalized and analyzed. Section 3 details a more sophisticated method of generating mixed power mean value theorems, by a limited adaptation of the iterative method itself. The form of these estimates offers many advantages over those of section 2, and provides the key to the elimination of the 16th power from (1.1). These mean value theorems are then applied to the proof of Theorem 1 in section 4. The tools developed here are applicable to a wide range of mixed power representation problems, and we briefly illustrate in section 5 the application to the problem of determining the number of terms required to represent all large n, when the lowest power used is a kth power instead of a square. Throughout, n is a large natural number whose representation as a sum of mixed powers is at issue, and E is an arbitrarily small positive real number. Constants implied by the Landau Oand Vinogradov <<-symbols may depend on E or k. For a real number x, write e(x) for e27iX, [x] for the greatest integer not exceeding x and llxll for the distance from x to the nearest integer. Unless otherwise specified, lowercase Latin letters denote natural numbers and Greek letters denote real numbers. Let A(P, R) denote the set of natural numbers not exceeding P with no

There are many continuous quantitative dimensions in the physical world. Philosophical, psychological, and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time and mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, and density) and number can be conceptualized discretely or continuously (i.e., natural vs real numbers). Variation on these quantitative dimensions is typically correlated, e.g., larger objects often weigh more than smaller ones. Number is a distinctive continuous dimension because the natural numbers (i.e., positive integers) are used to quantify collections of discrete objects. This aspect of number is emphasized by teaching of the count word sequence and arithmetic during the early school years. We review research on spatial and numerical estimation, and argue that a generalized magnitude system is the starting point for development in both domains. Development occurs along several lines: (1) changes in capacity, durability, and precision, (2) differentiation of the generalized magnitude system into separable dimensions, (3) formation of a discrete number system, i.e., the positive integers, (4) mapping the positive integers onto the continuous number line, and (5) acquiring abstract knowledge of the relations between pairs of systems. We discuss implications of this approach for teaching various topics in mathematics, including scaling, measurement, proportional reasoning, and fractions.

In this article we examine certain formulae applying to the Riemann zeta function in the critical strip 0<Re(s)<1, and by means of the Laplace transform we demonstrate new relations between existing formulae. Müntz has proposed a formula for the Riemann zeta function in the critical strip, which involves an arbitrary function satisfying certain conditions. From this formula it is clear that a resolution of the Riemann hypothesis may hinge on a successful method for dealing with the sum-integral difference. The Euler–MacLaurin summation formula is one such device. Here, we generate new expressions for certain Laplace transforms involving the sum-integral difference. Subsequently, the formulae thus far established are generalized using an arbitrary positive number m, which makes apparent the dependence of classical analysis on formulae and results associated with the exponential function which is characterized by the particular case m=1.

Let F(x) be a real function with sufficiently many derivatives existing and satisfying certain non-vanishing conditions in the interval [1,2]. Then ∑ m = M 2 M - 1 e ( T F ( m M ) ) = O ( M 1 / 2 T 9 / 56 log T ) as M, T → ∞ with T ⩾ M. Such sums often occur in the study of the zeta function. This result extends recent work of Bombieri and Iwaniec, whose ‘discrete Hardy–Littlewood method’ uses rational approximations to the derivatives of F(x) at rational points, Gauss sums, Poisson summation, and the large sieve.

We defined the quadratic Hermitian forms on a space of even dimension over a finite field; they are constructed from classical Hermitian forms. We give the exact value of exponential sums associated to the quadratic Hermitian forms. On the other hand, we define with the space of quadratic Hermitian forms a new family of subcodes of the Reed-Muller code of order two, and we give the value of the parameters of these codes and of the associated trace codes.