Numbers In Arithmetic Progressions Research Articles

Двоїстий алгоритм пошуку простих чисел на відрізках великих розмірностей

A matrix model of subsequence of natural numbers with a multiplicative basis from the first prime numbers is proposed. The matrix model is a square matrix, where vector columns are arithmetic progressions with difference and number of progressions equal to product of base elements. By crossing out arithmetic progressions with first terms which are multiples of base elements, we obtain a symmetric sparse matrix that contains all the prime numbers of subsequence of natural numbers, except for the base ones, which increases the density of prime numbers in sparse matrices. Sparse matrices are not formed clearly, only the vector of first terms of arithmetic progressions is formed. Properties of sparse matrices have been proven. Formulas that speed up the calculation of compound numbers in arithmetic progressions have been derived, the structures of vector elements of first terms of arithmetic progressions have been determined, the connectivity of symmetric parts of sparse matrices has been investigated. With the expansion of the base the number of pairs of elements with the difference equal to the power of two («twins», «fours», etc.) increases in the vector of first terms. This is a necessary condition for the existence of constants for which linear equations of two variables can have an infinite set of solutions in prime numbers. The irregularity of distribution of prime numbers in subsequences of natural numbers is related to the structure of elements of the vector of first terms. An algorithm for finding prime numbers on segments of large dimensions with a parallel calculation process has been built. The proposed algorithm is binary to algorithms for sifting subsequences of natural numbers by prime divisors. In these algorithms, it is not possible to parallelize the calculation process, since screening procedure requires storing the numerical information from the preceding steps (vector model of array processing). The binary algorithm calculates compound numbers in each pair of arithmetic progressions with symmetric first terms simultaneously, using only vector of the first terms of arithmetic progressions, which makes possible processing of large-dimensional arrays

The Infinitude of Primitive Abundant Numbers in Arithmetic Progressions

Summary Using Dirichlet’s theorem on arithmetic progressions, we show that if a and b are positive integers with a deficient greatest common divisor, the arithmetic progression ak + b contains infinitely many primitive abundant numbers.

Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection

In this paper, we study moments of Hurwitz class numbers associated to imaginary quadratic orders restricted into fixed arithmetic progressions. In particular, we fix t t in an arithmetic progression t ≡ m ( mod M ) t\equiv m\ \, \left ( \operatorname {mod} \, M \right ) and consider the ratio of the 2 k 2k -th moment to the zeroeth moment for H ( 4 n − t 2 ) H(4n-t^2) as one varies n n . The special case n = p r n=p^r yields as a consequence asymptotic formulas for moments of the trace t ≡ m ( mod M ) t\equiv m\ \, \left ( \operatorname {mod} \, M \right ) of Frobenius on elliptic curves over finite fields with p r p^r elements.

Formulas for moments of class numbers in arithmetic progressions

We obtain explicit formulas for the second moments of Hurwitz class numbers $H(4n-t^2)$ with $t$ running through a fixed congruence class modulo $3$.

Whole Numbers in Specified Arrays and Their Relationships in Multi – Dimensional Locales

The study details specified properties of whole numbers in conjunction with repetitive arrays and sequences. There prevails a common pattern for numbers when they exist in defined structures. The paper extends to the scope of progressions in regard to the specific number relationships and its reach in advanced mathematical studies. The properties of numbers enumerated have its scope in the field of recreational mathematical theories as well. Finding 01 proves the relationships between numbers in specific patterns, with exceptions in effect. The exceptions hold true only in the mentioned patterns of numbers. Findings 02 and 03 show the properties of numbers in Arithmetic Progression and the relationship of the numbers in specific arrangements with the Common Difference. Findings 04 and 05 work on the value of Determinants of matrices of which numbers are in Arithmetic and Geometric Progressions and its application in regard to positioning of multiple points in varying dimensions. Findings 06, 07 and 08 detail the unique properties of numbers when applied in addition, subtraction and multiplication and the ‘CONSTANCY’ in the end results. Findings 09 and 10 explain the relationships and properties of numbers with reference to the structures of regular polygons and the positioning of points in multiple dimensions.

A variance fork-free numbers in arithmetic progressions of given modulus

An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes (see reference [3]). We improve one of the error terms.

Number of arithmetic progressions in dense random subsets of ℤ/nℤ

Number of arithmetic progressions in dense random subsets of ℤ/nℤ

A Note on Cube-Full Numbers in Arithmetic Progression

We obtain an asymptotic formula for the cube-full numbers in an arithmetic progression n ≡ l mod q , where q , l = 1 . By extending the construction derived from Dirichlet’s hyperbola method and relying on Kloosterman-type exponential sum method, we improve the very recent error term with x 118 / 4029 < q .

AN AVERAGE THEOREM FOR TUPLES OF k ‐FREE NUMBERS IN ARITHMETIC PROGRESSIONS

We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.

Average distribution of k‐free numbers in arithmetic progressions

AbstractWe obtain a new bound on the average value of the error term in the asymptotic formula for the number of k‐free numbers in arithmetic progressions. In particular, we improve the results of J. Gibson (2014) and C. Hooley (1975).

On the computation of identities relating partition numbers in arithmetic progressions with eta quotients: An implementation of Radu's algorithm

In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg, in which the generating functions of a partition function over a given set of arithmetic progressions are expressed in terms of Dedekind eta quotients over a given congruence subgroup. These identities include the famous results by Ramanujan which provide a witness to the divisibility properties of p(5n+4), p(7n+5). We give an implementation of this algorithm using Mathematica. The basic theory is first described, and an outline of the algorithm is briefly given, in order to describe the functionality and utility of our package. We thereafter give multiple examples of applications to recent work in partition theory. In many cases we have used our package to derive alternate proofs of various identities or congruences; in other cases we have improved previously established identities.

On smooth square‐free numbers in arithmetic progressions

A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M. Z. Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$ and also consider more general versions of this question replacing $p$-smoothness of $s$ by the stronger condition of $p^{\alpha}$-smoothness. Using bounds on multiplicative character sums and a sieve method, we also show that we can represent all residue classes by a positive square-free integer $s\le p^{2+o(1)}$ which is $p^{1/(4e^{ /2})+o(1)}$-smooth. Additionally, we obtain stronger results for almost all primes $p$.

О линейной независимости значений некоторых гипергеометрических функций над мнимым квадратичным полем

Основная трудность, с которой приходится иметь дело при исследовании арифметической природы значений обобщенных гипергеометрических функций с иррациональными параметрами, состоит в том, что общий наименьший знаменатель нескольких первых коэффициентов соответствующих степенных рядов растет слишком быстро с увеличением числа этих коэффициентов. Последнее обстоятельство делает невозможным использование известного в теории трансцендентных чисел метода Зигеля для проведения упомянутого исследования. Применение названного метода предполагает использование принципа Дирихле для построения функциональной линейной приближающей формы. Это построение является первым этапом длинного и сложного рассуждения, приводящего в конечном итоге к получению требуемого арифметического результата. Попытка применить принцип Дирихле в случае функций с иррациональными параметрами наталкивается на непреодолимые трудности из-за упомянутого выше слишком быстрого роста общего наименьшего знаменателя коэффициентов соответствующих рядов Тейлора. Вследствие этого в случае функций с иррациональными параметрами обычно применяют эффективное построение линейной приближающей формы (или совокупности таких форм при использовании совместных приближений). Коэффициенты построенной формы являются многочленами с алгебраическими коэффициентами. Для общего наименьшего знаменателя этих коэффициентов требуется затем получить приемлемую оценку сверху его абсолютной величины. Известные оценки такого рода нуждаются в некоторых случаях в уточнении. Это уточнение осуществляется с применением теории делимости в квадратичных полях; дополнительно используются сведения о распределении простых чисел в арифметической прогрессии. В настоящей работе рассматривается один из вариантов эффективного построения совместных приближений для гипергеометрической функции общего вида и ее производных. Общий наименьший знаменатель коэффициентов многочленов, входящих в эти приближения, оценивается затем с помощью уточненного варианта соответствующей леммы. Все это позволяет получить новый результат об арифметической природе значений указанной функции в малой по абсолютной величине ненулевой точке мнимого квадратичного поля.

Means of some arithmetical functions on shifted smooth numbers in arithmetic progression

We provide asymptotic estimates for sums of some arithmetical functions taken over shifted smooth numbers in arithmetic progression related to the sum of digits function. Our results are based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to G Tenenbaum, Stephanie S Loiperdinger and Igor E Shparlinski.

Klaus Friedrich Roth, 1925-2015

Klaus Friedrich Roth, 1925-2015

On Prime Values of Some Quadratic Polynomials

The problem on the prime values of polynomials in one variable with rational integral coefficients has been solved, up to now, only for the polynomials of degree one by the famous Dirichlet theorem on prime numbers in arithmetic progressions. In this paper, we start studying properties of prime numbers represented by certain polynomials of degree two. Bibliography: 2 titles.

Squarefull numbers in arithmetic progression II

In this paper, we improve the error term in a previous paper on an asymptotic formula for the number of squarefull numbers in an arithmetic progression.

Squarefree numbers in arithmetic progressions

We give asymptotics for correlation sums linked with the distribution of squarefree numbers in arithmetic progressions over a fixed modulus. As a particular case we improve previous results concerning the variance.

On Bourgain's bound for short exponential sums and squarefree numbers

We use Bourgain's recent bound for short exponential sums to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.

Character sums over squarefree and squarefull numbers

We give upper bounds for character sums over squarefree and squarefull numbers sharper than the prior known in the literature. As an application, we study the distribution of squarefull numbers in arithmetic progressions and sharpen (without restriction on the modulus) the recent results obtained by Liu and Zhang.