All Questions
35 questions
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
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
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
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
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 ...
- 179
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 ...
- 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Unpublished result of Rosser in Sieve Methods book
Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert.
...
- 24.8k
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
-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
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
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 ...
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
$$
\sum_{\substack{1 \leq n \leq X,\\ p \nmid n \text{ for all }p \in \mathcal{P}}} 1....
- 2,207
1
vote
1
answer
356
views
Some questions about some examples in "sieve methods" in the book "Opera de Cribro" by Friedlander and Iwaniec
I am reading the book "Opera de Cribro - John B. Friedlander, Henryk Iwaniec" and in pages 5,6 I do not understand why and how they chose $X$, $A(x)$, $A_d(x)$, $g(p)$ and $r_d(x)$.
any hints will be ...
- 841
6
votes
0
answers
233
views
admissible tuples vs. prime tuples
Let $\rho^\ast(x)$ denote the maximal length of an admissible sequence in $[1,x]$, i.e. of a sequence which does not cover all the residue classes modulo any $n\geq 2$. Hensley and Richards (1974) ...
- 105k
3
votes
1
answer
356
views
Squarefree values of polynomials at prime arguments
This is a reference request.
Assume that $f_1,\ldots,f_r \in \mathbb{Z}[t]$ are non-zero linear polynomial.
Letting $\mu$ be the M\"{o}bius function, is there any work on
$$ \sum_{p\leq x} \prod_{i=...
- 3,062
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
$$\delta_y := \limsup_{x \...
user40023
1
vote
1
answer
203
views
Best bound on $p, p+2k$ with $k$ fixed
Given some integer $k>0$, there are $O(x/\log^2 x)$ primes $p \le x$ such that $p+2k$ is also prime. It has been conjectured at least since Hardy-Littlewood that
$$
\pi_{2k}(x) \sim c_{2k}\int_2^x\...
- 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 ...
- 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 ...
- 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 ...
- 2,283
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 ...
- 2,576
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 ...
- 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 ...
- 2,576
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 ...
- 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) $
...
7
votes
1
answer
431
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}+...
- 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
$$
\mathfrak{S}(d_1, \ldots, ...
2
votes
0
answers
175
views
Best known Upper bound on Twin Primes [duplicate]
I know that there is a result from J Wu that the number of twin primes less than a given magnitude $N$ does not exceed
$$\frac{2aCN}{\log^2{N}}$$
Where $C=\prod \frac{p(p-2)}{(p-1)^2}$ and $a$ is ...
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 ...
- 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 ...
- 2,719