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

Similarity:

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

Similarity:

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

Similarity:

On the Brun-Titchmarsh theorem

James Maynard (2013)

Acta Arithmetica

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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.