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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
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
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 $...
Jens Reinhold's user avatar
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
0 votes
0 answers
462 views

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 ...
user140242's user avatar
3 votes
1 answer
228 views

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 ...
Charles's user avatar
  • 9,114
14 votes
1 answer
424 views

Unpublished result of Rosser in Sieve Methods book

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. ...
Bjørn Kjos-Hanssen'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
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
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
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 ...
tomos's user avatar
  • 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 < \...
user152169's user avatar
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 $S$ ...
Vincent Granville's user avatar
-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 ...
user267839's user avatar
  • 6,028
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
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
7 votes
1 answer
343 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....
Kurisuto Asutora's user avatar
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 ...
asad's user avatar
  • 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) ...
GH from MO's user avatar
  • 105k
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
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 \...
user avatar
1 vote
1 answer
203 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\...
Charles's user avatar
  • 9,114
2 votes
0 answers
197 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 ...
H A Helfgott's user avatar
  • 20.2k
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
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
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
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
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
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
8 votes
1 answer
910 views

Is this weak asymptotic Goldbach's conjecture open?

Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes. Goldbach's conjecture: $\tau(x) = 0$ Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $ ...
Sebastien Palcoux's user avatar
7 votes
1 answer
430 views

Are primes of density 0 in $a\cdot b^n+c$?

Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms $n\cdot2^{n+a}+...
Charles's user avatar
  • 9,114
12 votes
0 answers
627 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
2 votes
0 answers
175 views

Best known Upper bound on Twin Primes [duplicate]

I know that there is a result from J Wu that the number of twin primes less than a given magnitude $N$ does not exceed $$\frac{2aCN}{\log^2{N}}$$ Where $C=\prod \frac{p(p-2)}{(p-1)^2}$ and $a$ is ...
Alexander Botros'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
24 votes
3 answers
2k views

Are sets with similar asymptotic behavior as the primes necessarily finite additive bases?

The set of primes $\mathbb{P}$ has many interesting properties in additive number theory and some of the most famous open problems about $\mathbb{P}$ are the well-known Goldbach's strong and weak ...
Joni Teräväinen's user avatar