The arithmetic-progression tag has no wiki summary.

**4**

votes

**1**answer

248 views

### non-asymptotic Bertrand-type theorems for arithmetic progression

It is well known that primes of form $4k+3$, call them $3=q_1 < q_2 < \dots$ satisfy $q_{n+1}/q_n\rightarrow 1$ (and even $q_n=\frac{n}{2\log n}(1+o(1))$). I would be glad to see results of ...

**12**

votes

**2**answers

577 views

### Arithmetic progressions modulo $p$ under the squaring map

I feel that the following problem should be known, but I'm not sure where to look for it.
Fix a real constant $\frac{1}{2} \ge \epsilon > 0$. For varying primes $p$, Let $A_p$ denote the set of ...

**9**

votes

**2**answers

698 views

### What are the analytic properties of Dirichlet Euler products restricted to arithmetic progressions?

There are (at least) two ways of writing down the Dirichlet L-function associated to a given character χ: as a Dirichlet series
$$\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$
or as an Euler product
...

**8**

votes

**2**answers

487 views

### Primes in quasi-arithmetic progressions?

Suppose $\alpha > 1$ is irrational. Are there infinitely many primes of the form $\left\lfloor \alpha n \right\rfloor$? Is the number of $p \leq X$ of this form $\sim \alpha^{-1} X ...

**3**

votes

**1**answer

992 views

### Strengthening of Dirichlet's theorem on arithmetic progressions

Hello all, Dirichlet's famous theorem asserts that any arithmetic progression
$\lbrace ax+b | x \in {\mathbb N}\rbrace$ contains infinitely many primes if a
and b are relatively prime.
I am ...

**6**

votes

**1**answer

701 views

### Primes in arithmetic progressions

Denote by $\pi(x,a,q)$ the number of primes $p\le x$ of the form $p=qk+a$
and $E(x,a,q)=\phi(q)^{-1}\mathrm{Li}(x)-\pi(x,a,q)$.
What is the strongest conjectured bound on $E(x,a,q)$ in terms of $x,q$?
...

**0**

votes

**1**answer

292 views

### Do there exist sets of integers with arbitrarily large upper density which contains infinitely many elements that are not in an arithmetic progression of length 3?

Given that simply stipulating positive upper density is not sufficient to guarantee that all but finitely many members are in an arithmetic progression of length 3, that there indeed exists sets of ...

**5**

votes

**2**answers

611 views

### Inverse Length 3 Arithmetic Progression Problem for sets with positive upper density

It is a famous theorem of Roth, which Szemerédi famously generalized, that if a set of natural numbers has positive upper density then it contains arithmetic progressions of length $k$. The famous ...

**15**

votes

**3**answers

1k views

### What is the shortest route to Roth's theorem?

Roth first proved that any subset of the integers with positive density contains a three term arithmetic progression in 1953. Since then, many other proofs have emerged (I can think of eight off the ...

**16**

votes

**1**answer

777 views

### Covering the primes by arithmetic progressions

Define the length of a set of arithmetic progressions
of natural numbers
$A=\lbrace A_1, A_2, \ldots \rbrace$
to be $\min_i | A_i |$: the length of the shortest sequence
among all the progressions.
...

**11**

votes

**3**answers

726 views

### What does the computer suggest about the parity of p(n), for n in a fixed arithmetic progression?

Let p(n) be the number of partitions of n. A famous theorem of Euler allows one to compute
the parity of p(n) quickly for quite large n. In:
On the distribution of parity in the partition function, ...

**35**

votes

**1**answer

8k views

### Is the Green-Tao theorem true for primes within a given arithmetic progression?

Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.
Now, consider an arithmetic progression with starting term $a$ and common difference $d$. ...

**17**

votes

**5**answers

2k views

### Finitely many arithmetic progressions

A few years ago, somebody told me a lovely problem. I suspect there may be more to it (which I would be interested in learning), and would very much like to find a reference, it makes me uncomfortable ...

**3**

votes

**2**answers

461 views

### “half arithmetic progressions” in dense sets

Fix a positive real number d>0. Szemeredi's theorem implies that for every integer k, there exists an integer N(k,d) such that if A is a subset of the interval [1,N] with density greater than d >0, ...

**5**

votes

**2**answers

404 views

### Smallest k-term AP of primes

Let $S(k)$ denote the smallest integer such that there exists a k-term arithmetic progression of primes among the integers $[1,S(k)]$. Green and Tao have an unpublished note that gives a very large ...

**9**

votes

**1**answer

552 views

### Covering the primes by 3-term APs ?

Hello, the Green-Tao theorem says infinitely many k-term Arithmetic Progressions exist for any integer k.
My question is: can we actually partition the primes into 3-term APs only (or is there a ...

**7**

votes

**1**answer

599 views

### Upper bound for number of k-term arithmetic progressions in the primes

Normal heuristics give that number of k-term arithmetic progressions in [1,N] should be about
\[c_k\frac{N^2}{\log^kN}\]
for some constant $c_k$ dependent on k. ...