Questions tagged [sieve-theory]

Question 1

I'm following the Worked Example for the Special Number Field Sieve paper and I got stuck at the Schirokauer Map construction. Could someone verify my map construction logic and help me answer these ...

Question 2

Is it possible to use the sieve method to solve problems like these? Count a number of $p_1p_2\cdots p_r$, a product of $r\geqslant 1$ primes such that $p_i\in [P,2P]$ for all $i=1,2,\ldots, r$ in ...

Question 3

In the chapter on sieve methods, specifically the $\Lambda^{2}$-sieve, Iwaniec-Kowalski state that $$\sum_{d < \sqrt{D}}^{\flat} h(d) > \prod_{p | q} (1-g(p)) \log \sqrt{D},$$ where $h(d) = \...

Question 4

Let $x$ be large and $$ A = \{1,3,5,\dots,\le x\} $$ be the odd integers $\le x$. For each odd prime $p \le x$ and for each integer $k$, remove from $A$ all integers $$ n \equiv \frac{p-9}{2} \pmod p \...

Question 5

Let $x$ be large and consider the odd integers $$ A = \{1,3,5,\dots,\le x\}. $$ For each odd prime $p \ge 3$ with $p \le x$, choose two residue classes $a_p, b_p \pmod p$. Assume that for every such $...

Question 6

I have been experimenting with a family of Eratosthenes-like sieves that seem to generate long streaks of primes before producing the first composite. I would like to ask if something like this is ...

Question 7

Following up on my previous question, suppose we fix parameters $\theta \in (0, 1/2)$ and $\eta > 0$. For each large $X$, an adversary chooses a set $\mathcal{R}(X)$ of residue classes of the form $...

Question 8

Fix a large parameter $X$. For $0<\delta<\theta<\tfrac12$, set $H := X^{\theta}$. Question. Is it true that there exist infinitely many values of $X$ and, for each such $X$, a modulus $q\le X^...

Question 9

Consider the equilateral triangular (Eisenstein) lattice $L = \mathbb{Z}\langle a, b \rangle \subset \mathbb{R}^2$ with $a = (1, 0)$ and $b = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. A wrap is ...

Question 10

Fix an even integer $h \geq 12$ and set $W = \prod_{p < h} p$. Choose a residue class $b \pmod{W}$ with the covering property that for every $s$ in the range $1 \leq s \leq h-1$, there is a prime $...

Question 11

By the prime number theorem, we have $\frac{\pi(n)}n \sim \frac1{\log(n)}$. On the other hand, I was wondering whether the asymptotic behavior of $\frac{\pi(n)}n$ can be figured out by a naive sieving,...

Question 12

I'm studying the Jurkat-Richert theorem from Nathanson's text "Additive Number Theory - the classical bases, 1996" and I think there is an error at a crucial point. The equality from page ...

Question 13

Let $\pi (x+y)-\pi (x)$ be the number of primes in $(x,x+y)$ and define $$A_d=\sum _{x<n\leq x+y\atop {d|n}}1\hspace {10mm}\Psi _z=\sum _{x<n\leq x+y\atop {p|n\implies p>z}}1.$$ The following ...

Question 14

Disclaimer: as I'm not sure whether this question is research-level or not, I ask it here rather than on Mathoverflow. In sieve theory, one aims at sieving a set $S$ of integers through modular ...

Question 15

Let $\lambda^{+} : \mathbb{N} \to \mathbb{R}$ be an upper bound sifting function, i.e., $\lambda^+$ satisfy the inequality $$ \sum_{d | n} \lambda^+(d) \geq \sum_{d | n} \mu(d) = \begin{cases} 1 &...

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Questions tagged [sieve-theory]

Computing the Schirokauer map in Number Field Sieve for Discrete Logarithms

Sieve method with primes in the selected range

An inequality in Iwaniec-Kowalski

Is $S_k(x)$ non-empty for all $k$ for sufficiently large $x$?

Can two (nonzero) forbidden residues per odd prime $p \ge 3$ ever remove all odd integers $\le x$?

A scalable sieve generating long prime streaks

Are bounded prime gaps robust to adversarial residue-class deletion?

Simultaneous prime-free short intervals modulo small $q$

Almost-prime lap counts in primitive hexagonal wraps

Single-AP two-form dispersion beyond $1/2$ on short-interval averages?

Density of prime numbers based on (naive) sieving [duplicate]

Substitution inside an integral: an error in Nathanson's proof of the Jurkat-Richert theorem?

Getting Chebyshev for an interval with sieves

Affine symmetry of a sieved set

Inequality involving sifting functions

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