All Questions
Tagged with analytic-number-theory sieve-theory
31 questions with no upvoted or accepted answers
12
votes
0
answers
628
views
Sieve bound for prime $k$-tuples
Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by
$$
\mathfrak{S}(d_1, \ldots, ...
10
votes
0
answers
350
views
Are there are any attempts utilising sieve theory to attack the general $a p \pm 1$ problem?
It is currently an open question if there are infinitely many primes $p$ such that $2p + 1$ is prime (Sophie Germain primes) or that at least one of $24p \pm 1$ is prime.
Could Zhang's method, or the ...
- 533
6
votes
0
answers
233
views
admissible tuples vs. prime tuples
Let $\rho^\ast(x)$ denote the maximal length of an admissible sequence in $[1,x]$, i.e. of a sequence which does not cover all the residue classes modulo any $n\geq 2$. Hensley and Richards (1974) ...
- 105k
6
votes
0
answers
296
views
Constants for Rosser's Sieve
I am trying to apply Iwaniec's formulation of Rosser's sieve (here) to obtain nontrivial lower bounds for almost-primes in various sequences. These sequences have sieve dimension 1 (if $g(p)$ is the ...
- 1,161
6
votes
0
answers
505
views
$x^2+1$ attaining almost prime values
Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...
- 3,062
5
votes
0
answers
326
views
Counting primes, twin primes, cousin primes: unusual approach, connection to some conjectures
I am investigating the following sieve-like algorithm. Let $S_N=\{1,\dots,N\}$. For all primes $p$ with $p_0\leq p \leq M$, we remove from $S_N$ the following elements: all numbers $n\in S_N$ such ...
- 3,098
5
votes
0
answers
130
views
On Ford's "The distribution of integers with a divisor in a given interval"
Let $H(x,y,z)$ denote the number of positive integers of size at most $x$ which have a divisor in the range $(y,z]$. In his famous "The distribution of integers with a divisor in a given interval&...
- 2,207
5
votes
0
answers
408
views
Paging Henryk Iwaniec: Problems In Lemma 1?
I gritted my teeth and dove into some sieve theory. In his 1978 article On the Problem of Jacobsthal in Demonstratio Mathematica, Iwaniec presents two Lemmas to prove his main result (leading to $j(n)...
- 13k
4
votes
0
answers
174
views
Counting number of primes $p$ less than $x$ with certain constraints on prime divisors of $p-1$ for Artin's Conjecture on primitive root
I was reading this paper. There is a Lemma 1 saying that for a fixed prime $q$ the cardinality of the set $\{p \leq x \mid (\frac{q}{p})=-1\}$ such that all odd prime divisors of $p-1$ are greater ...
- 521
4
votes
0
answers
163
views
Large Gaps Between Almost Primes
What is the best lower bound for the longest interval contained in $[1,x]$ free of primes and products of two primes? In other words I am asking for the best lower bounds in a variant of the ...
- 2,283
3
votes
0
answers
76
views
Divisor of given order in short intervals
Is the following Open question or Conjecture already known, or eventually settled ?
Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
- 433
3
votes
0
answers
232
views
Numbers made up of primes from a given set
Take a set $\mathcal P$ of primes and denote by $\langle \mathcal P\rangle $ the set of all natural numbers composed of primes from $\mathcal P$. If
\[ \sum _{p\in \mathcal P}\frac {1}{p}\]
converges ...
- 1,381
3
votes
0
answers
252
views
Counting twin primes with a sieve-like algorithm
The sequence A002822, denoted as $S$, represents all the twin primes except $\{3, 5\}$. Other than that exception, $k$ and $k+2$ are twin primes iff $(k+1)/6\in S$. Let $S(N)$ be the subset of $S$ ...
- 3,098
3
votes
0
answers
173
views
Kubilius model in higher sieve dimension?
The Kubilius model, based on the fundamental lemma of sieve theory, let us approximate the probability of events depending on the variables $X_p$, $p\leq y$, where $X_p=1$ if $p|n$ ($n$ a random ...
- 20.2k
3
votes
0
answers
106
views
Friedlander-Iwaniec Flipping moduli
I am reading section 12 (Flipping Moduli) of the paper "The polynomial $X^2+Y^4$ captures its primes" by Friedlander and Iwaniec.
At page 997, just below equation (12.7) we start estimating the ...
- 31
3
votes
0
answers
205
views
Sum of multiplicative arithmetic function over squarefree numbers
In the "Sieve methods" notes of Dimitris Koukoulopoulos (see http://www.dms.umontreal.ca/~koukoulo/documents/notes/sievemethods.pdf), the following useful result can be found:
Theorem 0.4.1. Let $g$ ...
user40023
3
votes
0
answers
97
views
A sieve result with two parameters
I proved the following sieve result and - since the proof is quite long and I need to use it in a work - I am looking for a reference to it (or at least something from which it could be proved quickly)...
user40023
3
votes
0
answers
603
views
Ramanujan conjecture and covariance of Kloosterman sums
There has been interest in moments and covariances/correlations of Kloosterman sums $S(m,n,c)=\sum_{ad=1\ (\text{mod}\ c)} e(\frac{ma+nd}{c})$ like
$\sum_{m\in\mathbb F_c} S(m,n,c)^k$, $\sum\sum_{m_1,...
- 982
2
votes
1
answer
227
views
Sieve Method works for variant question?
There are multiple results on the sieve method, and I wanted to ask about the following variant
(to know if it is trivial by one of the current versions of the sieve method, or seems a challenging ...
- 21
2
votes
0
answers
179
views
A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound
In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$
I ...
user344548
2
votes
0
answers
119
views
Almost-prime values attained by polynomials, with extra conditions
Given integers $a_1,\ldots,a_k>0$ and $b_1,\ldots,b_k$, consider the polynomial $f(x) = \prod_{i=1}^{k} (a_i x +b_i) \in \mathbb{Z}[x]$. Suppose that $\{ a_i x+b_i\}_{i=1}^{k}$ are pairwise ...
- 14.6k
2
votes
0
answers
120
views
Sieving the values of an arithmetic sequence which is infinitely many times $1$
I have a sequence of positive integers $a_n$ which assumes infinitely many times the value $1$. I want to estimate the cardinality of the following set:
$$\#\{n\leq x : a_n>1 \text{ and } (a_n, \...
2
votes
0
answers
189
views
Lattice Sieving in Number Field Sieve
I am currently going through Pollard's article on Lattice Sieving and have a few confusions. Firstly, how to figure that $C$ and $D$ in the two-dimensional array so that every $(c,d)$ pair corresponds ...
- 63
1
vote
0
answers
192
views
Prerequisites for Chen's theorem?
I am an undergraduate theoretical physics student, and I am trying to understand Chen's theorem. But when I tried to read Chen Jingrun's 1973 paper (https://www.sciengine.com/Math%20A0/doi/10.1360/...
- 11
1
vote
0
answers
95
views
Sieve theory obstruction: prime-sparse and nearly full-differenced sets?
Let $D(A) = {|a-b| : a, b \in A}$ denote the difference set of $A \subseteq \mathbb{Z}$. A set $A \subseteq (x/2, x]$ is almost full-differenced if $|D(A)| \geq \frac{x}{2} - \log x$. Let $C_x$ denote ...
1
vote
0
answers
148
views
Counting prime factors of polynomial functions
Let $\Omega(n)$ denote the number of prime factors (counted with multiplicity) of a non-zero integer $n$. For $f \in \mathbb Z[X]$ non-zero, let $$m(f) = \liminf_{n \to \infty} \Omega(f(n))$$
(1) Is $...
- 11.9k
1
vote
0
answers
94
views
Large sieve inequality-like sum without the square
Let $S(\alpha) = \sum_{n\leq N} w(n) e^{2\pi i \alpha n}$ for some function $w$ defined on $\mathbb{R}$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for ...
- 579
1
vote
0
answers
106
views
Sieve Theory uniform bound in Richert's Lectures on Sieves
I'm not sure how suitable this question is, but I have had no response on Mathematics Stack Exchange. My original question is here: https://math.stackexchange.com/questions/3402938/uniformity-...
- 111
1
vote
0
answers
139
views
Sieving question
How many integers $n\leq X$ are there with the property that $\prod_{p\in S} p \geq n^{1/2-\epsilon}$? Here (to keep notation readable) I've written $p\in S$ if and only if $p||n$ (that is, $p|n$ and $...
- 793
0
votes
0
answers
54
views
On the upper bound estimation of $D(N)$ in Chen Jingrun's theorem
What are the current research results on the estimation of the upper bound of $D(N)$ in Chen Jingrun's theorem?
Including but not limited to Chen Jingrun's improvement 7.8342 and Wu Jie's improvement ...
- 1
0
votes
0
answers
183
views
Prime counting function estimate sieve of Eratosthenes-Legendre
I'm trying to arrive at estimate 1.17 (page 21) of Koukoulopoulos lecture notes [https://dms.umontreal.ca/~koukoulo/documents/notes/sievemethods.pdf]
$$\#\{n \leq x : p|n \Rightarrow p > \sqrt{x}\}...
