On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II

Giovanni Coppola; Saverio Salerno

Displaying similar documents to “On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II”

On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev (1997)

Journal de théorie des nombres de Bordeaux

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In this paper we consider the asymptotic formula for the number of the solutions of the equation $p_{1} + p_{2} + p_{3} = N$ where $N$ is an odd integer and the unknowns $p_{i}$ are prime numbers of the form $p_{i} = [n^{1 / γ_{i}}]$ . We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case $γ_{1} = γ_{2} = γ_{3} = γ$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $γ$ for $50 / 53$ < $γ$ < $1$ . ...

On the distribution of $p^{α}$ modulo one

Xiaodong Cao, Wenguang Zhai (1999)

Journal de théorie des nombres de Bordeaux

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In this paper, we give a new upper-bound for the discrepancy $D (N) : = sup_{0 \leq γ \leq 0} | \sum_{\binom{p \leq / N}{\{p^{α}\} \leq γ}} 1 - π (N) γ |$ for the sequence $(p^{α})$ , when $5 / 3 \leq α > 3$ and $α \neq 2$ .

Absolutely continuous distribution functions of additive functions $f (p) = {(l o g p)}^{- a}, a > 0$

G. Jogesh Babu (1975)

Acta Arithmetica

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On the Brun-Titchmarsh theorem

James Maynard (2013)

Acta Arithmetica

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The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size $x / (q^{1 / 2} ϕ (q))$ when logx/logq ≥ 8 and...

On the powerful part of $n^{2} + 1$

Jan-Christoph Puchta (2003)

Archivum Mathematicum

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We show that $n^{2} + 1$ is powerfull for $O (x^{2 / 5 + ϵ})$ integers $n \leq x$ at most, thus answering a question of P. Ribenboim.

Note on special arithmetic and geometric means

Horst Alzer (1994)

Commentationes Mathematicae Universitatis Carolinae

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We prove: If $A (n)$ and $G (n)$ denote the arithmetic and geometric means of the first $n$ positive integers, then the sequence $n \mapsto n A (n) / G (n) - (n - 1) A (n - 1) / G (n - 1)$ $(n \geq 2)$ is strictly increasing and converges to $e / 2$ , as $n$ tends to $\infty$ .

Automaticity IV : sequences, sets, and diversity

Jeffrey Shallit (1996)

Journal de théorie des nombres de Bordeaux

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This paper studies the descriptional complexity of (i) sequences over a finite alphabet ; and (ii) subsets of $N$ (the natural numbers). If ${(s (i))}_{i \geq 0}$ is a sequence over a finite alphabet $Δ$ , then we define the $k$ - of $s, A_{s}^{k} (n)$ , to be the smallest possible number of states in any deterministic finite automaton that, for all $i$ with $0 \leq i \leq n$ , takes $i$ expressed in base $k$ as input and computes $s (i)$ . We give examples of sequences that have high automaticity in all bases $k$ ; for example, we show that the characteristic...

The binary Goldbach conjecture with primes in arithmetic progressions with large modulus

Claus Bauer, Yonghui Wang (2013)

Acta Arithmetica

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It is proved that for almost all prime numbers $k \leq N^{1 / 4 - ϵ}$ , any fixed integer b₂, (b₂,k) = 1, and almost all integers b₁, 1 ≤ b₁ ≤ k, (b₁,k) = 1, almost all integers n satisfying n ≡ b₁ + b₂ (mod k) can be written as the sum of two primes p₁ and p₂ satisfying $p_{i} \equiv b_{i} (m o d k)$ , i = 1,2. For the proof of this result, new estimates for exponential sums over primes in arithmetic progressions are derived.

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

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It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer $N_{r}$ such that the polynomial $f (X) / N_{r}$ represents at least r distinct primes.

On the range of Carmichael's universal-exponent function

Florian Luca, Carl Pomerance (2014)

Acta Arithmetica

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Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds $x / {(l o g x)}^{. 36}$ for all large x, while for φ it is equal to $x / {(l o g x)}^{1 + o (1)}$ , an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.

Sums of positive density subsets of the primes

Kaisa Matomäki (2013)

Acta Arithmetica

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We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1 - o (1)) α / (e^{γ} l o g l o g (1 / β))$ , which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $ℤ *_{m}$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A, B \subseteq ℤ *_{m}$ of densities α...

The value of additive forms at prime arguments

Roger J. Cook (2001)

Journal de théorie des nombres de Bordeaux

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Let $f (𝐩)$ be an additive form of degree $k$ with $s$ prime variables $p_{1}, p_{2}, \dots, p_{s}$ . Suppose that $f$ has real coefficients $λ_{i}$ with at least one ratio $λ_{i} / λ_{j}$ algebraic and irrational. If s is large enough then $f$ takes values close to almost all members of any well-spaced sequence. This complements earlier work of Brüdern, Cook and Perelli (linear forms) and Cook and Fox (quadratic forms). The result is based on Hua’s Lemma and, for $k \geq 6$ , Heath-Brown’s improvement on Hua’s Lemma.