All Questions
72 questions
6
votes
1
answer
292
views
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
$$
M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le ...
- 178
2
votes
1
answer
226
views
Sieve Method works for variant question?
There are multiple results on the sieve method, and I wanted to ask about the following variant
(to know if it is trivial by one of the current versions of the sieve method, or seems a challenging ...
- 21
7
votes
1
answer
276
views
From $\Lambda_k$ and $\Lambda$ to $\mu$ (or $\lambda$)
Let $\{a_n\}_{n=1}^\infty$, $a_n \in \mathbb{C}$, $|a_n|\leq 1$. Let $\Lambda_k = \mu \ast \log^k$; in particular, $\Lambda_1$ equals the von Mangoldt function $\Lambda$. Suppose that we have ...
- 20.2k
7
votes
4
answers
793
views
Must bounded sequences be well-distributed to most *composite* moduli?
Let $\{a_n\}_{n=1}^N$, $|a_n|\leq 1$. Let $Q=\sqrt{N}$. Then $a_n$ is well-distributed modulo most prime $p\leq Q$, in the following sense:
$$\sum_{p\leq Q} \frac{1}{p} \left(\frac{1}{N/p} \sum_{\...
- 20.2k
5
votes
1
answer
750
views
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
5
votes
2
answers
651
views
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
0
votes
1
answer
222
views
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
4
votes
2
answers
479
views
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 $$h>\...
- 45
2
votes
0
answers
179
views
A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound
In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$
I ...
user344548
4
votes
1
answer
286
views
Density of primes $p$ where $p-1$ has a prime factor exceeding $p^{2/3}$
Fouvry proved* that primes $p$ such that the greatest prime factor, $q$, of $p-1$ is greater than $p^{2/3}$ have positive density in the primes. (The sequence is A073024 in the OEIS.)
Are there any ...
- 9,114
1
vote
1
answer
244
views
Large sieve type inequality
Let $S_x(t)=\sum_{n\le x} a_n e(nt)$, where $e(x)=e^{2\pi i x}$. Then, the large sieve inequality tells us that
$$
\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
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
4
votes
2
answers
257
views
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 $y$):
$$L(Q)=\sum_{\substack{\delta_1,\delta_2\leq Q\\\delta_1\equiv0\ (a)\\\delta_2\equiv0\ (b)}}\frac{...
- 41
3
votes
0
answers
76
views
Divisor of given order in short intervals
Is the following Open question or Conjecture already known, or eventually settled ?
Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
- 433
3
votes
1
answer
293
views
Best available bounds for $\pi(Y)-\pi(Y-X)$?
I don't know much (anything) about sieves, but as I read the section on the Selberg upper bound sieve from Greaves's Sieves in Number Theory, there is a theorem 4 which says that
If $Y\ge X \ge 2$, ...
- 107
1
vote
0
answers
94
views
Large sieve inequality-like sum without the square
Let $S(\alpha) = \sum_{n\leq N} w(n) e^{2\pi i \alpha n}$ for some function $w$ defined on $\mathbb{R}$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for ...
- 579
9
votes
1
answer
825
views
Status of current research in Sieve Theory
I have done a course in Sieve Theory from the notes of Prof. Rudnick. Before this, I did 2 courses in Number Theory from the 2 volumes of Apostol.
I don't have any guidance by professor as I am living ...
- 793
5
votes
2
answers
701
views
Specific application of Cauchy-Schwarz and Large Sieve
Im reading a paper by Matomaki here, and the following is stated (I'm paraphrasing):
"By the Cauchy-Schwarz inequality and the large sieve, we have
$$\sum_{q \leq Q}\frac{q}{\phi(q)}\sum_{\...
- 153
5
votes
1
answer
214
views
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
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
3
votes
2
answers
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
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
13
votes
1
answer
777
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 < \...
- 253
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
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
6
votes
1
answer
636
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 ...
- 559
11
votes
2
answers
744
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 $1 \leq N \leq q$. ...
- 1,990
-1
votes
1
answer
258
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 ...
- 6,028
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
1
vote
1
answer
265
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 ...
- 263
11
votes
2
answers
1k
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 ...
- 2,075
-3
votes
1
answer
380
views
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 ...
- 793
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]
$$\#\{n \leq x : p|n \Rightarrow p > \sqrt{x}\}...
3
votes
1
answer
416
views
Primes with given Hamming weight
If I understand correctly, in the following thread
Are There Primes of Every Hamming Weight?
two users of the site claim that it has been already proven that, for every sufficiently large $n \in \...
- 87
15
votes
1
answer
1k
views
Why do Maynard-Tao weights succeed?
I'm attempting to understand why the Maynard-Tao weights are successful in proving bounded gaps between primes, but the GPY weights are not.
These two posts do an excellent job in giving an overview ...
3
votes
2
answers
386
views
Sieve bound for the sum of two squares
Let $$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) \le \sum_{p \le n}1 \ll \frac{n}{\log n}...
- 130
8
votes
1
answer
1k
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 ...
- 2,207
3
votes
1
answer
686
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 ...
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:
$$\#\{n\leq x : a_n>1 \text{ and } (a_n, \...
1
vote
1
answer
356
views
Some questions about some examples in "sieve methods" in the book "Opera de Cribro" by Friedlander and Iwaniec
I am reading the book "Opera de Cribro - John B. Friedlander, Henryk Iwaniec" and in pages 5,6 I do not understand why and how they chose $X$, $A(x)$, $A_d(x)$, $g(p)$ and $r_d(x)$.
any hints will be ...
- 841
6
votes
0
answers
233
views
admissible tuples vs. prime tuples
Let $\rho^\ast(x)$ denote the maximal length of an admissible sequence in $[1,x]$, i.e. of a sequence which does not cover all the residue classes modulo any $n\geq 2$. Hensley and Richards (1974) ...
- 105k
7
votes
1
answer
488
views
Examples of the large sieve inequality where a constant larger than 1 is needed
Let $S(x) = \sum_{n=0}^{N-1} a_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that
$$ \sum_{r=1}^R |S(x_r)|^2 ...
- 13k
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
3
votes
1
answer
356
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
$$ \sum_{p\leq x} \prod_{i=...
- 3,062
5
votes
1
answer
224
views
Locating a certain result on primes represented by a certain polynomial
In Theorem 2 of the paper "A polynomial divisor problem" by Friedlander and Iwaniec, Theorem 2 states that $$\sum_{a^6 + b^2\le x} \Lambda(a^6 + b^2)\sim cx^{2/3}$$ for some constant $c > 0$ (in ...
- 1,890
4
votes
1
answer
233
views
Shifted primes avoiding a set of divisors
Let $B$ be a set of positive integers such that $\sum_{b \in B} 1 / \varphi(b) < +\infty$, where $\varphi(\cdot)$ is the Euler's totient function. For any $y > 0$ put
$$\delta_y := \limsup_{x \...
user40023
3
votes
1
answer
528
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 ...
- 26.9k
2
votes
1
answer
171
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, ...
user40023
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