# Questions tagged [sieve-theory]

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Kubilius model in higher sieve dimension?

The Kubilius model, based on the fundamental lemma of sieve theory, let us approximate the probability of events depending on the variables $X_p$, $p\leq y$, where $X_p=1$ if $p|n$ ($n$ a random ...
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### 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 ...
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### Prime factors of $p-1$

It is one of the consequences of Sieve theory is that number of primes $p\leq x$ such that all prime divisors of $p-1$ are greater than $p^{\varepsilon},$ is $\gg \frac{x}{\log^2x}.$ In particular, ...
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### References of research papers which lead to starting of Sieve Theory

A bit of background in number theory which I have studied -> I have studied number theory from David M Burton's Elementary Number Theory and Apostol's Introduction to Analytic Number Theory and ...
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### Sieve Theory uniform bound in Richert's Lectures on Sieves

I'm not sure how suitable this question is, but I have had no response on Mathematics Stack Exchange. My original question is here: https://math.stackexchange.com/questions/3402938/uniformity-...
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### 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 ...
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### Almost-prime values attained by polynomials, with extra conditions

Given integers $a_1,\ldots,a_k>0$ and $b_1,\ldots,b_k$, consider the polynomial $f(x) = \prod_{i=1}^{k} (a_i x +b_i) \in \mathbb{Z}[x]$. Suppose that $\{ a_i x+b_i\}_{i=1}^{k}$ are pairwise ...
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### Sieving beyond threshold

This is a follow-up to the question here: Sum of divisors below threshold. User "Lucia" gave an excellent answer there, and probably the question below is very closely related. Still, since I am not ...
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### Sieving modulo non-prime residue classes

Let $n$ be a positive integer, and consider the set $\{1, \dots, n\}$. If we remove from this set all the numbers $a$ which satisfy $$a \equiv 0 \mod d$$ for at least one divisor $d$ of $n$ (...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### Lattice Sieving

What are some good references for Lattice Sieving in Number Field Sieve? Could someone suggest some research papers in this area?(Theoretical and Computational Perspective)
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### 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 ...
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### Sum of multiplicative arithmetic function over squarefree numbers

In the "Sieve methods" notes of Dimitris Koukoulopoulos (see http://www.dms.umontreal.ca/~koukoulo/documents/notes/sievemethods.pdf), the following useful result can be found: Theorem 0.4.1. Let $g$ ...
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### A sieve result with two parameters

I proved the following sieve result and - since the proof is quite long and I need to use it in a work - I am looking for a reference to it (or at least something from which it could be proved quickly)...
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### A sieve with two parameters

I am in need of a (relatively) general sieve with two parameters $y, z$. I am quite sure that on the literature there must be some result of the kind that I have in mind, probably a corollary of the ...
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### 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 ...
### 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 ...