Questions tagged [sieve-theory]
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139 questions
14
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Was Fermat's Last Theorem known for infinitely many primes before Wiles?
Before Andrew Wiles's 1997 proof of Fermat's Last Theorem, in 1985, Étienne Fouvry et al. proved that the first case of FLT holds for infinitely many primes $p$.
Is there any infinite class of primes ...
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Asymptotic viability of a symmetrical, prime-offset heuristic for Legendre's Conjecture [closed]
Recall that Legendre's Conjecture states that for every positive integer $n$, there is always a prime $p$ such that $n^2 < p < (n + 1)^2$.
I am currently working on a symmetrical approach to ...
8
votes
1
answer
213
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Proof for the de Bruijn–Erdős–Turán second moment asymptotic for $f(n)$
Consider the function $f(n) = \sum_{p<n}\frac{1}{n-p}$ where $p$ is prime. On Pages 62–63 of Problems and results on combinatorial number theory III, Erdős writes:
De Bruijn, Turán, and I ...
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4
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1
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Is this strengthening of the Maynard-Tao theorem on primes in admissible tuples known?
The groundbreaking work of Maynard and Tao showed the following fundamental result:
For any integer $m$, there exists a number $k(m)$ such that for any admissible set $H$ of $k$ integers (with $k$ at ...
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0
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0
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94
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Sieve method with primes in the selected range
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 ...
1
vote
0
answers
51
views
Positive proportion of one-sided large values for $\Re H(\sigma+it)$ short Dirichlet polynomial from Selberg sieve
Let $\sigma>\tfrac12$ be fixed and let $D\ge 2$ be a (large) parameter.
Take Selberg $\Lambda^2$-sieve weights $\{\lambda_d\}_{d\le D}$ supported on $d\le D$, with $\lambda_1=1$ and $\lvert\...
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2
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1
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Is $S_k(x)$ non-empty for all $k$ for sufficiently large $x$?
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
$$
n \equiv \frac{p-9}{2} \pmod p \...
- 23
0
votes
0
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68
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Maximal length of pairwise-coprime blocks in $\lfloor f(n) \rfloor$ for $C^2$ functions
Let $\mathcal{F}$ be the class of $C^2$ functions $f: [1, \infty) \to \mathbb{R}$ satisfying
$$f''(x) \to 0 \quad \text{and} \quad \limsup_{x \to \infty} |f'(x)| = \infty$$
Let $H(n)$ be a non-...
4
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0
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153
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Partitioning an infinite sumset into primes and composites
Let $A = \{a_1, a_2, \ldots\}$ and $B = \{b_1, b_2, \ldots\}$ be infinite, strictly increasing sequences of natural numbers. Define $S_{ij} = a_i + b_j$.
Question: Do there exist sequences $A$ and $B$ ...
0
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0
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53
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Minimal `s` for Non-emptiness of Sieved Sets with Two Forbidden Residues Modulo Primes
I am investigating a sieve theory problem concerning the non-emptiness of sets defined by forbidding two residues modulo each prime in a specific set. The setup is as follows:
Let $s$ be a positive ...
0
votes
1
answer
348
views
Euler product with asymptotic
In Polymath8b project there is that equation,
Which I do not understand the steps. I tried to fix a j and factorise,
$$\displaystyle S_j=\sum_{d_j,e_j}\frac{\mu(d_j)\mu(e_j)}{[d_j,e_j]{d_j}^s{e_j}^t} ...
- 45
1
vote
0
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172
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New local sieve for factorization and definition of primes within [x ^ 2, (x+1) ^ 2) [closed]
Good day!
I ask for your help – in investigating the twin prime conjecture, I am investigating a simple sawtooth function for the integer factorization of any natural number n = x ^ 2 + k in the range ...
1
vote
1
answer
242
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Continuous large sieve inequality
In Tao's blog we can find this exercise: Let $[M,M+N]$ be an interval for some $M \in {\bf R}$ and $N > 0$, and let $\xi_1,\dots,\xi_J \in {\bf R}$ be $\delta$-separated. For any complex numbers $...
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Sieving in the Gaussian integers
I asked this question as below a couple weeks ago in stackexchange but got no comments/answers, so I'll ask it here. (My understanding was that this is ok? Let me know if not).
Question: Can anyone ...
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7
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424
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Average of $\left(\frac{p}{q}\right)$ over $p\leq q^\epsilon$ for *most* $q$? (unconditionally)
Let $(\cdot/\cdot)$ be the Jacobi symbol. Consider the problem of showing that $(p/q)$ averages to $0$ as $p$ ranges over the primes $\leq q^\epsilon$, or some smaller range. For fixed $q$ (prime or ...
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