On Linnik's theorem on Goldbach numbers in short intervals and related problems

Alessandro Languasco; Alberto Perelli

Displaying similar documents to “On Linnik's theorem on Goldbach numbers in short intervals and related problems”

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 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...

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.

Trigonometric sums over primes III

Glyn Harman (2003)

Journal de théorie des nombres de Bordeaux

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New bounds are given for the exponential sum $\sum_{P \leq p < 2 P}^{} e (α p^{k})$ were $k \geq 5, p$ denotes a prime and $e (x) = exp (2 π i x)$ .

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

Giovanni Coppola, Saverio Salerno (2001)

Journal de théorie des nombres de Bordeaux

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This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials $(a n + b) (c n + d)$ in short intervals $n \in [x, x + x^{ϑ}]$ , where now $ϑ \in (0, 1]$ . Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level $ϑ$ , reaching $3 ϑ / 2$ (thus improving also the level of the previous paper, i.e. $3 ϑ - 3 / 2$ ), but our new results are different in structure. Then, we make a graphical comparison of the two methods.

Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

Zhao Feng (2017)

Open Mathematics

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In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., [...] N=p13+…+pj3 $\begin{matrix} N = p_{1}^{3} + ... + p_{j}^{3} \end{matrix}$ with [...] |pi−(N/j)1/3|≤N1/3−δ+ε(1≤i≤j), $\begin{matrix} | p_{i} - {(N / j)}^{1 / 3} | \leq N^{1 / 3 - δ + ε} (1 \leq i \leq j), \end{matrix}$ for some [...] 0<δ≤190. $\begin{matrix} 0 δ \leq \frac{1}{90} . \end{matrix}$ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

Consecutive primes in tuples

William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh (2015)

Acta Arithmetica

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In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple $(x) = {g x + h_{j}}_{j = 1}^{k}$ of linear forms in ℤ[x], the set $(n) = {g n + h_{j}}_{j = 1}^{k}$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $(n) = {g n + h_{j}}_{j = 1}^{k}$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps $δ_{1}, . . ., δ_{m}$ form an increasing...

Primes in almost all short intervals. II

Danilo Bazzanella (2000)

Bollettino dell'Unione Matematica Italiana

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In questo lavoro vengono migliorati i risultati ottenuti in «Primes in Almost All Short Intervals» riguardo la distribuzione dei primi in quasi tutti gli intervalli corti della forma $[g (n), g (n) + H]$ , con $g (n)$ funzione reale appartenente ad una ampia classe di funzioni. Il problema viene trattato mettendo in relazione l'insieme eccezionale per la distribuzione dei primi in intervalli nella forma $[g (n), g (n) + H]$ con l'insieme eccezionale per la formula asintotica $ψ (x + H) - ψ (x) \sim H as x \to \infty .$ I risultati presentati vengono quindi ottenuti grazie...

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...

Generalizing a theorem of Schur

Lenny Jones (2014)

Czechoslovak Mathematical Journal

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In a letter written to Landau in 1935, Schur stated that for any integer $t > 2$ , there are primes $p_{1} < p_{2} < \dots < p_{t}$ such that $p_{1} + p_{2} > p_{t}$ . In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers $t \geq k \geq 1$ and real $ϵ > 0$ , there exist primes $p_{1} < p_{2} < \dots < p_{t}$ such that $p_{1} + p_{2} + \dots + p_{k} > (k - ϵ) p_{t} .$

Carmichael numbers composed of primes from a Beatty sequence

William D. Banks, Aaron M. Yeager (2011)

Colloquium Mathematicae

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Let α,β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence $ℬ_{α, β} = {(⌊ α n + β ⌋)}_{n = 1}^{\infty}$ . We conjecture that the same result holds true when α is an irrational number of infinite type.