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**13**

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**1**answer

574 views

### Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $

A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale.
I would like ask about the much weaker statement forgetting ...

**10**

votes

**2**answers

626 views

### Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial

The starting point for this question is the following (false) statement
$\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$
Given a polynomial function $p:\mathbb{N} \to ...

**3**

votes

**0**answers

164 views

### Ramanujan conjecture and covariance of Kloosterman sums

There has been interest in moments and covariances/correlations of Kloosterman sums $S(m,n,c)=\sum_{ad=1\ (\text{mod}\ c)} e(\frac{ma+nd}{c})$ like
$\sum_{m\in\mathbb F_c} S(m,n,c)^k$, ...

**5**

votes

**0**answers

89 views

### Constants for Rosser's Sieve

I am trying to apply Iwaniec's formulation of Rosser's sieve (here) to obtain nontrivial lower bounds for almost-primes in various sequences. These sequences have sieve dimension 1 (if $g(p)$ is the ...

**1**

vote

**0**answers

75 views

### Sieving question

How many integers $n\leq X$ are there with the property that $\prod_{p\in S} p \geq n^{1/2-\epsilon}$? Here (to keep notation readable) I've written $p\in S$ if and only if $p||n$ (that is, $p|n$ and ...

**5**

votes

**0**answers

277 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 ...

**7**

votes

**1**answer

553 views

### Is this weak asymptotic Goldbach's conjecture open?

Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes.
Goldbach's conjecture: $\tau(x) = 0$
Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $
...

**2**

votes

**1**answer

364 views

### Proof of the Friedlander–Iwaniec theorem

Does anybody know where I could find the proof of the Friedlander–Iwaniec theorem. The link that I find when I search for it is http://www.pnas.org/content/94/4/1054.full.pdf+html, but this seems more ...

**0**

votes

**0**answers

206 views

### Relationship between this conjecture and Lehmer's Theorem?

Let A be:
n such that $\ \frac{n-1}{ord_n 2}=2^x\ $ and $n$ with the conditions of the conjecture in OEIS A226014,$\ n \in \mathbb{Z^+} ,\ x \in \mathbb{Z}_{\geq 0},\ $then $n$ is prime ...

**14**

votes

**2**answers

215 views

### S integral points of an elliptic curve, with S of positive density

Let E be an elliptic curve over Q of non-zero rank. Let S be the union of the primes of bad reduction of E with a Chebotarev set [1]. Suppose additionally that S has density strictly less than one.
...

**3**

votes

**1**answer

305 views

### An introduction to sieve method and their application, Cojocaru & Murty

On page 188.
Lemma 10.2.3 is $\sum_{\substack{\delta \leq x \\ 2 \nmid \delta}}\frac{\mu^{2}(\delta)}{\phi_{1}(\delta)} = A_{1}log(x) +A_{2} + O(\frac{1}{x^{1/4}})$ for positive $A_{1}$, $A_{2}$.
...

**10**

votes

**1**answer

389 views

### Does the Maynard-Tao Theorem apply to general tuples of linear forms?

In the paper http://arxiv.org/pdf/1311.5319v1.pdf the author states the following theorem, which he attributes to Maynard and Tao.
For any integer $m > 2$, there exists an integer
$k = k(m)$ such ...

**10**

votes

**2**answers

646 views

### What is a sieve and why are sieves useful?

I have been trying to understand what is exactly a sieve and why sieves are useful.
I have read Wikipedia articles about sieve theory but they don't provide a definition of what is a sieve or why they ...

**7**

votes

**1**answer

340 views

### Are primes of density 0 in $a\cdot b^n+c$?

Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms ...

**4**

votes

**0**answers

95 views

### On a weighted sum in a lemma for sieve methods

I'm reading James Maynard's paper "Small gaps between primes".
Lemma 6.1 (p.14) in this paper confused me. This lemma was taken from
Goldston-Graham-Pintz-Yildirim's paper "Small gaps between ...

**9**

votes

**0**answers

395 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, ...

**2**

votes

**0**answers

148 views

### Best known Upper bound on Twin Primes [duplicate]

I know that there is a result from J Wu that the number of twin primes less than a given magnitude $N$ does not exceed
$$\frac{2aCN}{\log^2{N}}$$
Where $C=\prod \frac{p(p-2)}{(p-1)^2}$ and $a$ is ...

**7**

votes

**1**answer

1k views

### A reformulation of the Riemann Hypothesis

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.
Let's define ...

**26**

votes

**1**answer

1k views

### Does a proof of Selberg's 3.2 inequality exist?

A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi ...

**11**

votes

**1**answer

734 views

### Least prime in an arithmetic progression and the Selberg sieve

My question concerns a technical step in the proof of Linnik's theorem on the least prime in an arithmetic progression, as presented in Chapter 18 of Iwaniec-Kowalski: Analytic number theory.
The ...

**5**

votes

**1**answer

230 views

### Large gaps between P2s

Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two ...

**23**

votes

**3**answers

907 views

### Are sets with similar asymptotic behavior as the primes necessarily finite additive bases?

The set of primes $\mathbb{P}$ has many interesting properties in additive number theory and some of the most famous open problems about $\mathbb{P}$ are the well-known Goldbach's strong and weak ...

**2**

votes

**1**answer

249 views

### 10 factors for x^2 coefficient in quadratic sieve?

I wrote a quadratic sieve and I tried plugging in all the same parameters as the wikipedia article says msieve uses: http://en.wikipedia.org/wiki/Quadratic_sieve#Parameters_from_realistic_example
It ...

**7**

votes

**1**answer

714 views

### Best possible sieves for the jacobsthal problem, linear programming, and the prime 2

Background/Motivation
Gerhard Paseman asked a question about bounds on the Jacobsthal function a while ago, which made me curious about whether the known bounds are best possible. Briefly, the ...

**16**

votes

**6**answers

3k views

### Erik Westzynthius's cool upper bound argument: update?

Version 2 of this writeup is
available, and includes a newer and simple upper bound thanks to
MathOverflow 88777 as
well as indirect references to future writeups. Details of further work
...