Questions tagged [sieve-theory]

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1answer
155 views

Least number coprime to a given integer

For a positive integer $n$ let $$f(n):=\min\{m\in \mathbb N: m>1, \gcd(m,n)=1\} .$$ Equivalently, $f(n) $ is the smallest prime not dividing $n$. Is there any upper bound literature for this? It is ...
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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 ...
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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 ...
13
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1answer
605 views

Large sieve inequality for sparse trigonometric polynomials

Let $S(\alpha) = \sum_{n\leq N}f(n) e^{2\pi i \alpha n}$ for some arithmetic function $f$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for some $0 < \...
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Order of magnitude on lower bounds for primes in intervals

I have been looking at the literature on sieve theory which proves theorems similar to the following: For all $x > x_0$ the interval $[x - x^{\theta}, x]$ contains prime numbers. For example, I ...
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198 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 $S$ ...
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108 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&...
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1answer
402 views

The history and original paper of the Rosser–Iwaniec sieve

I'm trying to find Rosser's original paper where he introduces his eponymous sieve. I've already found https://arxiv.org/pdf/math/0505521 (where the reference isn't given, but where it is indicated ...
11
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2answers
570 views

Improving the error term in a classic sieving problem

I'm new to asking questions on MathOverflow, so forgive me if this question is not the kind of thing to be asked here. Let $q$ be a positive integer and let $N$ be an integer with $1 \leq N \leq q$. ...
7
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1answer
190 views

How does the bound in the large sieve depend on the norm on the lattice?

I've been reading about the large sieve inequality in Serre's "Lectures on the Mordell-Weil theorem", which states it in the following setting, which I've simplified a bit here: Suppose $\...
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1answer
211 views

A number theoretical identity of exponential sum

I try to understand a number theoretical identity used by Jan-Christoph Schlage-Puchta in this answer. He defined the function $$S(\alpha)=\sum_{n\leq N}\Lambda(n) e(n\alpha)$$ where $\Lambda(n)$ is ...
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154 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 ...
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1answer
181 views

"Halfway" approach to Landau's 4th problem (special case of Bateman-Horn)

Landau's 4th problem asks if $n^2 + 1$ is prime for infinitely many $n \in \Bbb{Z}$. It is known that $n^2 + 1$ can only be divisible by Pythagorean primes, and that for any $p$ congruent to $1 \pmod ...
3
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1answer
132 views

Sieving by composite moduli

A traditional sieve gives a bound on the number of integers $n$ in an interval (say $I=[0,N]$) such that $$n\not\in S_p \mod p$$ for every prime $p$ in a set $\mathcal{P}$, where $S_p\subset \mathbb{Z}...
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840 views

What is the significance of Friedlander-Iwaniec and related theorems?

On p.177 of Number Theory Revealed: A Masterclass by Andrew Granville, the author states that "One can ask for prime values of polynomials in two or more variables." (though he later ...
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113 views

Prime factors of $p-1$

It is one of the consequences of Sieve theory is that number of primes $p\leq x$ such that all prime divisors of $p-1$ are greater than $p^{\varepsilon},$ is $\gg \frac{x}{\log^2x}.$ In particular, ...
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1answer
277 views

References of research papers which lead to starting of Sieve Theory

A bit of background in number theory which I have studied -> I have studied number theory from David M Burton's Elementary Number Theory and Apostol's Introduction to Analytic Number Theory and ...
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86 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-...
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139 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}\}...
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1answer
372 views

Primes with given Hamming weight

If I understand correctly, in the following thread Are There Primes of Every Hamming Weight? two users of the site claim that it has been already proven that, for every sufficiently large $n \in \...
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1answer
755 views

Why do Maynard-Tao weights succeed?

I'm attempting to understand why the Maynard-Tao weights are successful in proving bounded gaps between primes, but the GPY weights are not. These two posts do an excellent job in giving an overview ...
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101 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 ...
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1answer
176 views

Sieve bound for the sum of two squares

Let $$S(n) = \sum_{p \le n} b(n-p),$$ where $b(a)=1$ is $a$ is a sum of two squares of positive integers and $b(a)=0$ otherwise. Trivially by PNT we have $$S(n) \ll \sum_{p \le n}1 \le \frac{n}{\log n}...
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1answer
160 views

Almost-prime values attained by a product of quadratic polynomials

Let $F(x) = \prod_{i=1}^{k} (a_i x +b_i)$ be a product of $k$ linear polynomials, where $a_i,b_i$ are integers. Under very reasonable conditions, it is known that a constant $C_k$ exists with the ...
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113 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 ...
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169 views

Sieving beyond threshold

This is a follow-up to the question here: Sum of divisors below threshold. User "Lucia" gave an excellent answer there, and probably the question below is very closely related. Still, since I am not ...
2
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1answer
161 views

Sieving modulo non-prime residue classes

Let $n$ be a positive integer, and consider the set $\{1, \dots, n\}$. If we remove from this set all the numbers $a$ which satisfy $$ a \equiv 0 \mod d $$ for at least one divisor $d$ of $n$ (...
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1answer
605 views

Sum of divisors below threshold

Let $\sigma(n)$ denote the sum of divisors of $n$, that is, $$ \sigma(n) = \sum_{d | n} d. $$ It is known that $\sigma$ can have values as large as order $n \log \log n$. However, obviously the sum is ...
3
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1answer
478 views

Use of weights in the GPY's and Tao-Maynard's work on the twin prime conjecture

I am going through James Maynard's paper, Small Gaps between Primes, and have a number of questions regarding his approach. First, I am wondering why uses weights in his approach. While I generally ...
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108 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, \...
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232 views

Which continuous function is optimal for sieving?

In 1968, Barban and Vehov considered [1] the problem of determining for which continuous functions $\rho:\mathbb{R}^+\to [0,1]$ satisfying certain properties ($\rho(t)=1$ for $t\leq U_0$, $\rho(t)=0$ ...
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1answer
280 views

Proportion of numbers with prime divisors from restricted set

Let $X$ be large, and let $\mathcal{P} \subset \{1, \dots, X\}$ be a set of primes. What is a good upper bound for $$ \sum_{\substack{1 \leq n \leq X,\\ p \nmid n \text{ for all }p \in \mathcal{P}}} 1....
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1answer
281 views

Some questions about some examples in "sieve methods" in the book "Opera de Cribro" by Friedlander and Iwaniec

I am reading the book "Opera de Cribro - John B. Friedlander, Henryk Iwaniec" and in pages 5,6 I do not understand why and how they chose $X$, $A(x)$, $A_d(x)$, $g(p)$ and $r_d(x)$. any hints will be ...
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172 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) ...
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1answer
398 views

Examples of the large sieve inequality where a constant larger than 1 is needed

Let $S(x) = \sum_{n=0}^{N-1} a_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that $$ \sum_{r=1}^R |S(x_r)|^2 ...
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146 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 ...
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1answer
316 views

Squarefree values of polynomials at prime arguments

This is a reference request. Assume that $f_1,\ldots,f_r \in \mathbb{Z}[t]$ are non-zero linear polynomial. Letting $\mu$ be the M\"{o}bius function, is there any work on $$ \sum_{p\leq x} \prod_{i=...
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1answer
212 views

Lattice Sieving

What are some good references for Lattice Sieving in Number Field Sieve? Could someone suggest some research papers in this area?(Theoretical and Computational Perspective)
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1answer
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Locating a certain result on primes represented by a certain polynomial

In Theorem 2 of the paper "A polynomial divisor problem" by Friedlander and Iwaniec, Theorem 2 states that $$\sum_{a^6 + b^2\le x} \Lambda(a^6 + b^2)\sim cx^{2/3}$$ for some constant $c > 0$ (in ...
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1answer
172 views

Shifted primes avoiding a set of divisors

Let $B$ be a set of positive integers such that $\sum_{b \in B} 1 / \varphi(b) < +\infty$, where $\varphi(\cdot)$ is the Euler's totient function. For any $y > 0$ put $$\delta_y := \limsup_{x \...
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1answer
334 views

Higher dimensional large sieve inequality

One of the most important achievements in analytic number theory is the establishment of the so-called large sieve inequality, which is formulated as follows. Let $\{a_n\}$ denote a finite sequence of ...
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1answer
140 views

A sieve by not (necessarily) coprime integers

Let $\mathcal{A}$ be a nonempty set of pairwise coprime positive integers, and for each $a \in \mathcal{A}$ let $\Omega_a \subsetneq \{0, \ldots, a - 1\}$ be a set of residues modulo $a$. Furthermore, ...
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1answer
111 views

Bounding the number of polynomials whose Galois group is a subgroup of the alternating group

In the article of DAVID ZYWINA https://pdfs.semanticscholar.org/cd50/c2d3fb0ce0c6a66ee629419b69165b30d5bc.pdf. It says that using $n$- dimensional large sieve, we can get the bound |$\{$$ \ \ f(x)=x^...
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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 $g$ ...
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0answers
86 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)...
3
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1answer
154 views

A sieve with two parameters

I am in need of a (relatively) general sieve with two parameters $y, z$. I am quite sure that on the literature there must be some result of the kind that I have in mind, probably a corollary of the ...
1
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1answer
172 views

Best bound on $p, p+2k$ with $k$ fixed

Given some integer $k>0$, there are $O(x/\log^2 x)$ primes $p \le x$ such that $p+2k$ is also prime. It has been conjectured at least since Hardy-Littlewood that $$ \pi_{2k}(x) \sim c_{2k}\int_2^x\...
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189 views

Quasiprimes in arithmetic progressions

Let $$\Lambda_z(n) = \sum_{d|n, d>z} \mu(d) \log(d/z).$$ As S. Graham proved in 1978, $$\sum_{n\leq x} |\Lambda_z(n)|^2 \sim x \log(x/z).$$ provided $x\geq z$. We also know that, by the large ...
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168 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 ...
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1answer
345 views

Is the set of prime pairs such that $gcd(p−1,q−1)=2$ of positive density?

Is the set of prime pairs such that $gcd(p−1,q−1)=2$ of positive density? For example, for $p,q≤10^4$ the answer is approximately $1/2$. I was wondering if it were possible to use sieve methods and ...