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29 votes
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Why is there a Parity Problem in Sieve Theory and not a Mod p problem for any other p?

The "parity problem" in sieve theory, so far as I understand it, is the fact that sieves can't distinguish between primes and $2$-almost primes, numbers with exactly two prime factors, and will always ...
Xiaoyu He's user avatar
  • 1,161
16 votes
1 answer
1k views

Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $

A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale. I would like ask about the much weaker statement forgetting ...
john mangual's user avatar
  • 22.8k
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 ...
numbertheorylearner's user avatar
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 < \...
user152169's user avatar
12 votes
1 answer
1k views

Why do the Maynard-Tao weights work so well?

I am looking for an intuitive reason for why the Maynard-Tao weights work well to capture many primes of the form $n+h_1, \ldots , n+h_k$, where $(h_1, \ldots , h_k)$ is any admissible $k$-tuple. For ...
George Shakan's user avatar
12 votes
1 answer
1k views

Does the Maynard-Tao Theorem apply to general tuples of linear forms?

In the paper http://arxiv.org/pdf/1311.5319v1.pdf the author states the following theorem, which he attributes to Maynard and Tao. For any integer $m > 2$, there exists an integer $k = k(m)$ such ...
anon's user avatar
  • 303
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, ...
Jan-Christoph Schlage-Puchta's user avatar
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 ...
Favst's user avatar
  • 2,075
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 $1 \leq N \leq q$. ...
Joshua Stucky's user avatar
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 ...
KStar's user avatar
  • 533
9 votes
1 answer
403 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 ...
Gal Porat's user avatar
  • 225
9 votes
1 answer
826 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 ...
Arnold's user avatar
  • 793
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 ...
Kurisuto Asutora's user avatar
8 votes
1 answer
1k views

A reformulation of the Riemann Hypothesis

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes. Let's define $R(x,...
Subhajit Jana's user avatar
7 votes
2 answers
997 views

Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial

The starting point for this question is the following (false) statement $\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$ Given a polynomial function $p:\mathbb{N} \to \mathbb{N}$ ...
Dominic van der Zypen's user avatar
7 votes
4 answers
795 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_{\...
H A Helfgott's user avatar
  • 20.2k
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 ...
H A Helfgott's user avatar
  • 20.2k
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 ...
Mark Lewko's user avatar
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 ...
Cloudscape's user avatar
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 ...
Itachi's user avatar
  • 178
6 votes
2 answers
319 views

Evolution of partial sum of a sequence of induced Dirichlet characters

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$. Lets consider the sequence of induced characters $\chi^{P_N} $ obtained ...
Bertrand's user avatar
  • 1,199
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) ...
GH from MO's user avatar
  • 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 ...
Xiaoyu He's user avatar
  • 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 ...
Dr. Pi's user avatar
  • 3,062
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_{\...
CBagshaw's user avatar
  • 153
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-...
Nicky's user avatar
  • 365
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 ...
Krishnarjun's user avatar
5 votes
1 answer
455 views

Large gaps between P2s

Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two ...
Charles's user avatar
  • 9,114
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 ...
Mayank Pandey's user avatar
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 ...
Joshua Stucky's user avatar
5 votes
1 answer
189 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 ...
Ofir Gorodetsky's user avatar
5 votes
1 answer
1k views

Reference to "bounds of Weil and Deligne"

In the this paper by Friedlander and Iwaniec, it is said that they are "able to avoid much of the high-powered technology frequently used in modern analytic number theory such as the bounds of Weil ...
Mayank Pandey's user avatar
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 ...
Vincent Granville's user avatar
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&...
Kurisuto Asutora's user avatar
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)...
Gerhard Paseman's user avatar
4 votes
2 answers
840 views

Upper bound for the first Hardy-Littlewood conjecture

About the Hardy-Littlewood conjecture by Terence Tao: Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed ...
Alexey Milovanov's user avatar
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 ...
Charles's user avatar
  • 9,114
4 votes
1 answer
951 views

Number of twin primes

Consider number of twin primes less than $x$. We know that this number less than $\frac{Cx}{\log^2 x}$ for some constant $C$. Denote by $p_n$ the $n$-th prime number. Do we have the same result ...
Alexey Milovanov's user avatar
4 votes
1 answer
696 views

Relation between the binary Goldbach problem and binary version of Mobius sum

What I want to ask is about the structure of the Goldbach function that defined by $$ R(x)=\#\{ p \mid x-p \in \mathbb{P} , \ p\leq x/2\}$$ for $x\in 2\mathbb{N}$, where $\mathbb{P}$ is the set of ...
B . O's user avatar
  • 139
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{...
Tom Glover's user avatar
4 votes
1 answer
234 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 \...
user avatar
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>\...
Arda Yonet's user avatar
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 ...
dragoboy's user avatar
  • 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 ...
George Shakan's user avatar
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 ...
Dr. Pi's user avatar
  • 3,062
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 $Y\ge X \ge 2$, ...
user859588's user avatar
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=...
Dr. Pi's user avatar
  • 3,062
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 ...
Stanley Yao Xiao's user avatar
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 ...
Sultan Aitzhan's user avatar
3 votes
1 answer
216 views

Well-spacing of the roots of a quadratic congruence

On pages 956-957 of this paper, it is established that for any two $v_1, v_2$ satisfying $v_1^2 + 1 \equiv 0\operatorname{(mod} d_1), v_2^2 + 1\equiv 0\operatorname{(mod} d_2)$, $$\left\lVert \frac{...
Mayank Pandey's user avatar