Sign changes in $π_{q,a}(x) - π_{q,b}(x)$

Kevin Ford; Richard H. Hudson

Displaying similar documents to “Sign changes in $π_{q, a} (x) - π_{q, b} (x)$ ”

Chebyshev series expansions of the functions $J_{v} (k x) / {(k x)}^{v}$ and $I_{v} (k x) / {(k x)}^{v}$

Z. Cylkowski (1966)

Applicationes Mathematicae

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Chebyshev Distance

Roland Coghetto (2016)

Formalized Mathematics

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In [21], Marco Riccardi formalized that ℝN-basis n is a basis (in the algebraic sense defined in [26]) of [...] ℰTn $ℰ_{T}^{n}$ and in [20] he has formalized that [...] ℰTn $ℰ_{T}^{n}$ is second-countable, we build (in the topological sense defined in [23]) a denumerable base of [...] ℰTn $ℰ_{T}^{n}$ . Then we introduce the n-dimensional intervals (interval in n-dimensional Euclidean space, pavé (borné) de ℝn [16], semi-intervalle (borné) de ℝn [22]). We conclude with the definition of Chebyshev distance [11]. ...

Optimality of Chebyshev bounds for Beurling generalized numbers

Harold G. Diamond, Wen-Bin Zhang (2013)

Acta Arithmetica

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If the counting function N(x) of integers of a Beurling generalized number system satisfies both $\int_{1}^{\infty} x^{- 2} | N (x) - A x | d x < \infty$ and $x^{- 1} (l o g x) (N (x) - A x) = O (1)$ , then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that $\int_{1}^{\infty} | N (x) - A x | x^{- 2} d x < \infty$ and $x^{- 1} (l o g x) (N (x) - A x) = O (f (x))$ do not imply the Chebyshev bound.

Chebyshev polynomials and Pell equations over finite fields

Boaz Cohen (2021)

Czechoslovak Mathematical Journal

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We shall describe how to construct a fundamental solution for the Pell equation $x^{2} - m y^{2} = 1$ over finite fields of characteristic $p \neq 2$ . Especially, a complete description of the structure of these fundamental solutions will be given using Chebyshev polynomials. Furthermore, we shall describe the structure of the solutions of the general Pell equation $x^{2} - m y^{2} = n$ .

On some properties of Chebyshev polynomials

Hacène Belbachir, Farid Bencherif (2008)

Discussiones Mathematicae - General Algebra and Applications

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Letting $T_{n}$ (resp. $U_{n}$ ) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences ${(X^{k} T_{n - k})}_{k}$ and ${(X^{k} U_{n - k})}_{k}$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space $_{n} [X]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_{n}$ and $U_{n}$ admit remarkableness integer coordinates on each of the two basis.

The transfinite diameter of the real ball and simplex

T. Bloom, L. Bos, N. Levenberg (2012)

Annales Polonici Mathematici

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We calculate the transfinite diameter for the real unit ball $B_{d} : = x \in ℝ^{d} : | x | \leq 1$ and the real unit simplex $T_{d} : = x \in ℝ_{+}^{d} : \sum_{j = 1}^{d} x_{j} \leq 1 .$

Explicit extension maps in intersections of non-quasi-analytic classes

Jean Schmets, Manuel Valdivia (2005)

Annales Polonici Mathematici

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We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces $_{()} ({[- 1, 1]}^{r})$ ; (b) there is no continuous linear extension map from $Λ_{()}^{(r)}$ into $_{()} (ℝ^{r})$ ; (c) under some additional assumption on , there is an explicit extension map from $_{()} ({[- 1, 1]}^{r})$ into $_{()} ({[- 2, 2]}^{r})$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

Discriminants of Chebyshev radical extensions

T. Alden Gassert (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $t$ be any integer and fix an odd prime $ℓ$ . Let $Φ (x) = T_{ℓ}^{n} (x) - t$ denote the $n$ -fold composition of the Chebyshev polynomial of degree $ℓ$ shifted by $t$ . If this polynomial is irreducible, let $K = ℚ (θ)$ , where $θ$ is a root of $Φ$ . We use a theorem of Dedekind in conjunction with previous results of the author to give conditions on $t$ that ensure $K$ is monogenic. For other values of $t$ , we apply a result of Guàrdia, Montes, and Nart to obtain a formula for the discriminant of $K$ and compute an integral basis for the ring...

Renormings of $c_{0}$ and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this paper is to show that for every Banach space $(X, ∥ \cdot ∥)$ containing asymptotically isometric copy of the space $c_{0}$ there is a bounded, closed and convex set $C \subset X$ with the Chebyshev radius $r (C) = 1$ such that for every $k \geq 1$ there exists a $k$ -contractive mapping $T : C \to C$ with $∥ x - T x ∥ > 1 - 1 / k$ for any $x \in C$ .

An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

Časopis pro pěstování matematiky a fysiky

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A non-Chebyshev finite-dimensional subspace in $H_{1}$

S. B. Stechkin (1989)

Banach Center Publications

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On the proximate order of $f^{(s)} (z) * g^{(s)} (z)$ and ${[f (z) * g (z)]}^{(s)}$

M. K. Sen (1971)

Annales Polonici Mathematici

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The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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On C $^{*}$ -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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On a characterization of $L^{p}$ -norm

Janusz Matkowski (1989)

Annales Polonici Mathematici

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$ℵ_{1}$ -incompactness of Z

Ralph McKenzie (1971)

Colloquium Mathematicae

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On the $p^{λ}$ problem

Stephan Baier (2004)

Acta Arithmetica

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Localization techniques in $L^{p}$ spaces

A. Pełczyński, H. Rosenthal (1975)

Studia Mathematica

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Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula $π = \sum_{n = 0}^{\infty} ((5 n + 3) n! (2 n)!) / (2^{n - 1} (3 n + 2)!)$ with convergence as $13 . 5^{- n}$ , in much the same way as the Euler transformation gives $π = \sum_{n = 0}^{\infty} (2^{n + 1} n! n!) / (2 n + 1)!$ with convergence as $2^{- n}$ . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

А.М. Вершик (1972)

Zapiski naucnych seminarov Leningradskogo

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The general methods of finding the sum $S_{n}$ for all kinds of series, II

K. Orlov (1981)

Matematički Vesnik

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Displaying similar documents to “Sign changes in π q , a ( x ) - π q , b ( x ) ”

Displaying similar documents to “Sign changes in $π_{q, a} (x) - π_{q, b} (x)$ ”