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.

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.

A note on signs of Kloosterman sums

Kaisa Matomäki (2011)

Bulletin de la Société Mathématique de France

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We prove that the sign of Kloosterman sums $Kl (1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.