# Questions tagged [sieve-theory]

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### How much does one have to study connected fields to understand modern sieve methods? [closed]

For example, If I'd want to read through the "Primes in tuples" and other works on the GPY sieve, how much analysis/group theory/analytic number theory do I need to know?
• 101
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### 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 ...
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### Divisor of given order in short intervals

Is the following Open question or Conjecture already known, or eventually settled ? Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
• 388
1 vote
67 views

### Upper bound for the number of coprimes to primes below $x$ in an arbitrary interval of length $x$?

Let $\mathcal{E}$ be a subset of the primes up to $x^{{1/2}-o(1)}$ and let $S(T,T+x;\mathcal{E})$ be the number of integers in the interval $(T,T+x]$ that are coprime to the primes in $\mathcal{E}$. ...
• 955
278 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
432 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
1 vote
91 views

### Large sieve inequality-like sum without the square

Let $S(\alpha) = \sum_{n\leq N} w(n) e^{2\pi i \alpha n}$ for some function $w$ defined on $\mathbb{R}$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for ...
• 579
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### On the upper bound estimation of $D(N)$ in Chen Jingrun's theorem

What are the current research results on the estimation of the upper bound of $D(N)$ in Chen Jingrun's theorem? Including but not limited to Chen Jingrun's improvement 7.8342 and Wu Jie's improvement ...
190 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 ...
• 8,984
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### Status of current research in Sieve Theory

I have done a course in Sieve Theory from the notes of Prof. Rudnick. Before this, I did 2 courses in Number Theory from the 2 volumes of Apostol. I don't have any guidance by professor as I am living ...
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