# Questions tagged [sieve-theory]

The sieve-theory tag has no usage guidance.

112
questions

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### How much does one have to study connected fields to understand modern sieve methods? [closed]

For example, If I'd want to read through the "Primes in tuples" and other works on the GPY sieve, how much analysis/group theory/analytic number theory do I need to know?

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250
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### Density of primes $p$ where $p-1$ has a prime factor exceeding $p^{2/3}$

Fouvry proved* that primes $p$ such that the greatest prime factor, $q$, of $p-1$ is greater than $p^{2/3}$ have positive density in the primes. (The sequence is A073024 in the OEIS.)
Are there any ...

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1
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### Selberg sieve for counting monic irreducible $P \in \mathbb{F}_q[t]$ such that $P + K$ is also irreducible

In a 1983 paper by William Webb (link below), the author gives a version of the Selberg sieve for function fields and uses it to prove that for a fixed $K \in \mathbb{F}_q[t]$, the number $\mathcal{N}(...

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1
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### Large sieve type inequality

Let $S_x(t)=\sum_{n\le x} a_n e(nt)$, where $e(x)=e^{2\pi i x}$. Then, the large sieve inequality tells us that
$$
\sum_{q\le Q} \sum_{\substack{0\lt a \lt q \\ (a,q)=1}}|S_x(a/q)|^2 \le (Q^2+4\pi x)\...

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### 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 $...

4
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2
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251
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### Sum of $\frac{1}{(\delta_1,\delta_2)}$ with congruence restrictions

In the course of my work, I encountered the following sum ($(x,y)$ stands for the GCD of $x$ and $y$):
$$L(Q)=\sum_{\substack{\delta_1,\delta_2\leq Q\\\delta_1\equiv0\ (a)\\\delta_2\equiv0\ (b)}}\frac{...

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### Density of extended Mersenne numbers?

Consider the subset of odd positive integers defined and constructed as follows by these rules :
A) $1$ is in the set.
B) if $x$ is in the set , then $2x + 1$ is in the set.
C) if $x$ and $y$ are in ...

1
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1
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147
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### Functor whose essential image is a cosieve?

Definitions
An object $d \in Obj(\mathcal D)$ is in the essential image of $F$ if there exists some $c \in Obj(\mathcal C)$ such that $d \cong F c$.
A sieve in $\mathcal D$ is a full subcategory of $\...

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### 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 $[x,x+\log^2(x)]$ ...

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### Upper bound for the number of coprimes to primes below $x$ in an arbitrary interval of length $x$?

Let $\mathcal{E}$ be a subset of the primes up to $x^{{1/2}-o(1)}$ and let $S(T,T+x;\mathcal{E})$ be the number of integers in the interval $(T,T+x]$ that are coprime to the primes in $\mathcal{E}$. ...

3
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278
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### Best available bounds for $\pi(Y)-\pi(Y-X)$?

I don't know much (anything) about sieves, but as I read the section on the Selberg upper bound sieve from Greaves's Sieves in Number Theory, there is a theorem 4 which says that
If $Y\ge X \ge 2$, ...

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### Relation between sieve wheel and Sundaram sieve

I made this sieve for prime numbers, which I briefly describe:
We consider $\quad p=r+modulus \cdot k \quad$ with $\quad modulus=p_1*p_2* \cdots *p_m$
and then we choose an appropriate reduced ...

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### 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 ...

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### 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 ...

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### What fraction of the values of a quadratic polynomial can be prime?

I have an explicit, monic quadratic polynomial $P(x)$ and an integer $m$. Can I bound the number of prime values in $P(0), P(1), \ldots, P(m)$? A reference would be appreciated, if available. An ...

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### Status of current research in Sieve Theory

I have done a course in Sieve Theory from the notes of Prof. Rudnick. Before this, I did 2 courses in Number Theory from the 2 volumes of Apostol.
I don't have any guidance by professor as I am living ...

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### Specific application of Cauchy-Schwarz and Large Sieve

Im reading a paper by Matomaki here, and the following is stated (I'm paraphrasing):
"By the Cauchy-Schwarz inequality and the large sieve, we have
$$\sum_{q \leq Q}\frac{q}{\phi(q)}\sum_{\...

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1
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254
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### Sieve theory through variational principles

Disclaimer: I'm just starting to read Sieve Methods by Halberstam and Richert, so my present knowledge of the subject is close to zero, but it made me wonder if some connection to physics could exist, ...

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### Conjecture on a sieve of Flavius Josephus

Flavius Josephus's sieve: Start with the natural numbers; at the $k$-th sieving step, remove every $(k+1)$-st term of the sequence remaining after the $(k-1)$-st sieving step; iterate.
Some examples:
...

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402
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### Unpublished result of Rosser in Sieve Methods book

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert.
...

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### Remainder terms of congruence sums in sets of positive density

Let $\mathcal{A} \subset \mathbb{N}$ be an infinite sequence with positive density, in the sense that
$$
\tag{1}
\lim_{x\to\infty} \frac{|\mathcal{A} \cap x|}{x} = c > 0,
$$
and define the ...

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### 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 ...

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1
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278
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### 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 ...

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### 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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238
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### 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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### 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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### 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 ...

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### 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 $1 \leq N \leq q$. ...

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### 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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1
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237
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### 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 ...

3
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0
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### 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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1
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### "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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### 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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### 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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### 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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### References of research papers which lead to starting of Sieve Theory

Question - I am thinking to present one or two papers on Sieve Theory in my masters thesis. I will also present 3 other papers on Riemann Zeta Function which I have studied earlier . But I have no ...

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### 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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### 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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### 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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### 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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### 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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### 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) \le \sum_{p \le n}1 \ll \frac{n}{\log n}...

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### 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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### 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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### 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 ...

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### 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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### 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 ...