Questions tagged [sieve-theory]
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126 questions
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Complete list of functions known to satisfy the Siegel-Walfisz assumption
When I was reading Yitang Zhang's paper "Bounded Gaps between Primes" Text, on Page 1145, it is stated that "It should be remarked, by the Siegel-Walfisz theorem, that for all the
...
6
votes
1
answer
291
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Prime number theorem via large sieve type sums
We know that the prime number theorem is equivalent to the statement
$$
M(x)=\sum_{n\le x}\mu(n)=o(x).
$$
By using Ramanujan sums, we can write $M(x)$ as
$$
M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le ...
- 178
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0
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191
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Prerequisites for Chen's theorem?
I am an undergraduate theoretical physics student, and I am trying to understand Chen's theorem. But when I tried to read Chen Jingrun's 1973 paper (https://www.sciengine.com/Math%20A0/doi/10.1360/...
- 11
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226
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Sieve Method works for variant question?
There are multiple results on the sieve method, and I wanted to ask about the following variant
(to know if it is trivial by one of the current versions of the sieve method, or seems a challenging ...
- 21
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1
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276
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From $\Lambda_k$ and $\Lambda$ to $\mu$ (or $\lambda$)
Let $\{a_n\}_{n=1}^\infty$, $a_n \in \mathbb{C}$, $|a_n|\leq 1$. Let $\Lambda_k = \mu \ast \log^k$; in particular, $\Lambda_1$ equals the von Mangoldt function $\Lambda$. Suppose that we have ...
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4
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793
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Must bounded sequences be well-distributed to most *composite* moduli?
Let $\{a_n\}_{n=1}^N$, $|a_n|\leq 1$. Let $Q=\sqrt{N}$. Then $a_n$ is well-distributed modulo most prime $p\leq Q$, in the following sense:
$$\sum_{p\leq Q} \frac{1}{p} \left(\frac{1}{N/p} \sum_{\...
- 20.2k
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1
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748
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Sum of reciprocals of rough numbers
Let $x$ and $y$ be given real numbers. We may suppose that $2\leqslant x \leqslant y$ and that $u:= \log(y)/\log(x)$ remains bounded in a compact set away from $1$ as $x,y\to\infty$. An integer $n$ is ...
- 589
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95
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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 ...
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2
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650
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The twin prime problem and the Jurkat-Richert Theorem
Where does the Jurkat-Richert Theorem for linear sieves fail when applied to the twin prime problem?
I'm reading the last two chapters of Additive Number Theory The Classical Bases. The Jurkat-...
- 365
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221
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Trying to understand last part of the proof of normalized prime gap
We know that $$\liminf_{n\to\infty}{\frac{p_{n+1}-p_n}{\log p_n}}=0.$$
I'm trying to figure out the proof and I have read a lot of documents, I asked a question here. Still I can't see what's going on....
- 45
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2
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479
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Understanding the proof of Goldston–Pintz–Yíldírím's theorem
I hope this question fits the mission of this site.
In "Primes in Tuples I" theorem 2 says,
$$\liminf_{n\to\infty}{\frac{p_{n+1}-p_n}{\log p_n}}=0.$$
After a sieving progress you get $$h>\...
- 45
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1
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159
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Are there infinitely many primes $p$ such that $p +2$ has at most two distinct prime factors?
using lower bound sieve, one can show that there are infinitely many prime $p$ such that $p+2$ has at most four distinct prime factors [Theorem 10.2.1, 1].
Has there been any improvement of the above ...
- 365
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0
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179
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A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound
In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$
I ...
user344548
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Fundamental lemma of sieve theory in function fields
Is there any literature concerning the fundamental lemma of sieve theory in $\mathbb{F}_q[T]$?
In integers there are various versions of the lemma (bases on different sieves); I would be happy with ...
- 14.6k
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56
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How much does one have to study connected fields to understand modern sieve methods? [closed]
For example, If I'd want to read through the "Primes in tuples" and other works on the GPY sieve, how much analysis/group theory/analytic number theory do I need to know?
- 101
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285
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Density of primes $p$ where $p-1$ has a prime factor exceeding $p^{2/3}$
Fouvry proved* that primes $p$ such that the greatest prime factor, $q$, of $p-1$ is greater than $p^{2/3}$ have positive density in the primes. (The sequence is A073024 in the OEIS.)
Are there any ...
- 9,114
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169
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Selberg sieve for counting monic irreducible $P \in \mathbb{F}_q[t]$ such that $P + K$ is also irreducible
In a 1983 paper by William Webb (link below), the author gives a version of the Selberg sieve for function fields and uses it to prove that for a fixed $K \in \mathbb{F}_q[t]$, the number $\mathcal{N}(...
- 23
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1
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244
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Large sieve type inequality
Let $S_x(t)=\sum_{n\le x} a_n e(nt)$, where $e(x)=e^{2\pi i x}$. Then, the large sieve inequality tells us that
$$
\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)\...
- 178
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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 $...
- 11.9k
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257
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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{...
- 41
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339
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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 ...
- 769
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1
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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 $\...
- 511
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76
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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)]$ ...
- 423
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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}$. ...
- 965
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293
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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$, ...
- 107
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461
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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 ...
- 179
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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 ...
- 579
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54
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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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227
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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 ...
- 9,114
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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 ...
- 793
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2
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700
views
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_{\...
- 153
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1
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268
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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:
...
- 4,928
14
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1
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424
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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 ...
- 1,990
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350
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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 ...
- 533
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2
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465
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 ...
- 3,062
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232
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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 ...
- 1,381
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326
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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 ...
- 3,098
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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 < \...
- 253
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0
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70
views
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 ...
- 121
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252
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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 $S$ ...
- 3,098
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130
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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&...
- 2,197
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votes
1
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636
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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 ...
- 559
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2
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744
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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 $1 \leq N \leq q$. ...
- 1,990
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1
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225
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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 $\...
- 4,279
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1
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258
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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 ...
- 5,998
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0
answers
173
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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 ...
- 20.2k
1
vote
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 ...
- 263
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1
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179
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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}...
- 20.2k