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Showing 1 to 20 of 22 for “"arithmetic progressions"”.
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Explicit Estimates for Functions of Primes in Arithmetic Progressions
… the error term in the prime number theorem for arithmetic progressions, and Waring's problem for cubes.
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A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions
… in 1998 to prove the Prime Number Theorem for arithmetic progressions. We will review basic results from Dirichlet characters and L-functions. Furthermore, we establish a weak version of the Wiener-Ikehara Tauberian Theorem, which is an essential tool for the proof of our main result.
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Higher-order Fourier analysis with applications to additive combinatorics and theoretical computer science
… the primes contain infinitely-many three-term arithmetic progressions. Over the past two decades, a theory of higher-order Fourier analysis has been developed to study additive patterns which are not amenable to classical Fourier-analytic techniques. For example, while three-term arithmetic …
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Sparse regularity and relative Szemerédi theorems
… which states that there are arbitrarily long arithmetic progressions in the primes. The key step, known as a relative Szemerédi theorem, says that any positive proportion subset of a pseudorandom set of integers contains long arithmetic progressions. We give a simple proof of a strengthening …
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An Exposition Of Dirichlet’s Theorem
… the location and size of the prime numbers. Arithmetic progressions are very easily described subsets of the integers, yet they are infinite so it might be the case that they contain infinitely many prime numbers. Using Euclid’s original proof of the infiniteness of the primes as a model, we …
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Quadratic Reciprocity: Proofs and Applications
… cases of Dirichlet's theorem on primes in arithmetic progressions and Fermat's theorem on sums of two squares.
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The Factoradic Integers
The arithmetic progressions under addition and composition satisfy the usual rules of arithmetic with a modified distributive law. The basic algebra of such mathematical structures is examined; this leads to the consideration of the integers as a metric space under the "factoradic metric", i.e., …
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Some Extensions of the Skolem-Mahler-Lech Theorem
… these zero coefficients forms a finite union of arithmetic progressions modulo a finite set. Since rational functions satisfy linear differential equations of order 0 with polynomial coefficients, it is natural to conjecture that the Skolem-Mahler-Lech theorem also holds for functions satisfying …
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Sequências Numéricas: Aplicação de forma dinâmica.
… sequence, identifying regularities; studying the arithmetic progressions (PA), from its concept how to calculate and express the general term of an AP and the sum of its terms, and also conceptualize geometric progression(GP), to express and calculate the general term of a GP and the sum of its …
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Accent and Grouping Structures in the String Quartets of Béla Bartók
… In addition, symmetrical structures and arithmetic progressions are discovered. In many ways, Bartók's rhythmic organization mimics his procedures of pitch structuring.
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A Contribution to Metric Diophantine Approximation : the Lebesgue and Hausdorff Theories
… numerators and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This provides the first example of a Khintchine type result in the context of so–called uniform approximation.
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Extremal graph theory: supersaturation and enumeration
… of subsets of [n] which does not contain an arithmetic progression of a fixed length. This addresses another question of Cameron and Erdos and provides an optimal bound for infinitely many n. As corollaries, we improve the known transference results on arithmetic progressions.
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Formalising Combinatorial Structures and Proof Techniques in Isabelle/HOL
… Szemerédi's regularity lemma, Roth's theorem on arithmetic progressions, and the Balog-Szemerédi-Gowers theorem. Through this work, I aim to present a new approach to mathematical formalisation which focuses on developing general, modular formal proof techniques and modelling approaches, rather …
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Some topics in analytic and probabilistic number theory
… for the y-smooth numbers less than x among arithmetic progressions to modulus q, asymptotically as (logx)/(logq)-+ oo, subject to a certain condition on the relative sizes of y and q. The main point of this work is that it does not require any restrictions on the relative sizes of x and y. …
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Behavior of partition values modulo powers of primes
… is easy to define, questions on the arithmetic of its values lie much deeper. Ninety years ago, Ramanujan proved the celebrated congruences for <italic>p(n)</italic> which bear his name. For example, he proved, for all <italic>n</italic>, that <italic>p(5n + 4) == 0 mod 5</italic>. …
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Partition asymptotics; zeros of zeta functions; and Apéry-like numbers
PART I G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for the number of partitions into perfect kth powers, which was later proved by E. M. Wright. Recently, R. C. Vaughan …
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Food preferences and demand in the brushtail possum (Trichosurus vulpecula)
… between a geometric sequence (basis 2), and an arithmetic sequence (step 5) to ascertain if progression type affected the demand for foods. The food pairs of berries and egg, and two new foods of a barley and coco-pop® mix and rolled oats were tested. The same response rate patterns were …
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The distribution of k-free numbers and integers with fixed number of prime factors
Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms
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Vergleichende multiplikative Zahlentheorie
Unter der Annahme der verallgemeinerten Riemannschen Vermutung beweise ich eine effektive Variante des Satzes von Littlewood, partielle Ergebnisse zum Shanks-Renyi-race-Problem sowie Aussagen ueber lokale Schwankungen in der Primzahlverteilung. Als Hilfsmittel fuehre ich einen neuen Begriff von …
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Combinatorial Problems with Geometric Flavour
… A covering system is a finite collection of arithmetic progressions $\{a_1\text{ }(\text{mod } m_1),a_2\text{ }(\text{mod } m_2), \hdots, a_k\text{ }(\text{mod } m_k) \}$ that cover the integers, i.e., $\cup_i \{a_i+ n m_i \text{ : } n \in \mathbb{Z}\}=\mathbb{Z}$. Since their introduction by …
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