On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev

Displaying similar documents to “On the Piatetski-Shapiro-Vinogradov theorem”

Composite positive integers whose sum of prime factors is prime

Florian Luca, Damon Moodley (2020)

Archivum Mathematicum

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In this note, we show that the counting function of the number of composite positive integers $n \leq x$ such that $β (n) = \sum_{p ∣ n} p$ is a prime is of order of magnitude at least $x / {(log x)}^{3}$ and at most $x / log x$ .

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.

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...

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 $p_{1} + p_{2} + \dots + p_{k} > (k - ϵ) p_{t} .$

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...

Truncatable primes and unavoidable sets of divisors

Artūras Dubickas (2006)

Acta Mathematica Universitatis Ostraviensis

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We are interested whether there is a nonnegative integer $u_{0}$ and an infinite sequence of digits $u_{1}, u_{2}, u_{3}, \dots$ in base $b$ such that the numbers $u_{0} b^{n} + u_{1} b^{n - 1} + \dots + u_{n - 1} b + u_{n},$ where $n = 0, 1, 2, \dots,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S .$ If any such sequence contains infinitely many elements divisible by at least one prime number $p \in S,$ then we call the set $S$ unavoidable with respect to $b$ . It was proved earlier that unavoidable sets in base $b$ exist if $b \in {2, 3, 4, 6},$ and that no unavoidable set exists in base $b = 5 .$ Now,...

Infinitude of Wilson primes for $_{q} [t]$

Jim Sauerberg, Linghsueh Shu, Dinesh S. Thakur, George Todd (2013)

Acta Arithmetica

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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.

Prime numbers along Rudin–Shapiro sequences

Christian Mauduit, Joël Rivat (2015)

Journal of the European Mathematical Society

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For a large class of digital functions $f$ , we estimate the sums $\sum_{n \leq x} Λ (n) f (n)$ (and $\sum_{n \leq x} μ (n) f (n)$ , where $Λ$ denotes the von Mangoldt function (and $μ$ the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.

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.

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 a divisibility problem

Shichun Yang, Florian Luca, Alain Togbé (2019)

Mathematica Bohemica

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Let $p_{1}, p_{2}, \dots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k \geq 5$ , then $(p_{k + 1} - 1)! ∣ (\frac{1}{2} (p_{k + 1} - 1))! p_{k}!,$ which improves a previous result of the second author.

Shifted values of the largest prime factor function and its average value in short intervals

Jean-Marie De Koninck, Imre Kátai (2016)

Colloquium Mathematicae

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We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_{q} (n)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${(α s_{q} (P (n)))}_{n \in ℕ}$ is uniformly distributed modulo 1.