All Questions
9 questions with no upvoted or accepted answers
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, ...
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
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
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
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
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
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
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
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