# Tagged Questions

**7**

votes

**1**answer

258 views

### Prime races à la Mertens

I have just read the nice survey by Granville and Martin about prime races.
I wonder what happens if one changes the rules for the prime races as follows.
Fix $q$ a modulus (an integer $>1$). For ...

**9**

votes

**1**answer

361 views

### Roots of unity near 1 in Z / p Z

Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p.
The ...

**6**

votes

**0**answers

152 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 ...

**3**

votes

**1**answer

342 views

### When does Merten's product theorem accurately estimate the number of coprimes in an interval?

Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is:
Let $W(z)=\prod_{p\leq z}\left(1-\frac{1}{p}\right)$. For ...

**4**

votes

**1**answer

400 views

### Lines in image; are they significant to prime numbers if so how?

Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where ...

**10**

votes

**1**answer

400 views

### Chebotarev density theorem for $k$-almost primes

Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...

**4**

votes

**0**answers

284 views

### About sign changes of Li(x)-π(x)

Given a constant $C$, which are the best known upper bounds for the number of sign changes
of the function
$$
f: \mathbb{N} \rightarrow \mathbb{R}, \ \ x \mapsto {\rm Li}(x)-\pi(x)
$$
in the range ...

**0**

votes

**0**answers

84 views

### In this prime counting identity, can this limit of a sum be expressed as integrals?

Here's the identity I'm working with.
$E_{0}(n,b) = 1$
$E_{k}(n,b) = \displaystyle\sum_{j=2}^{\lfloor n \rfloor}E_{k-1}(\frac{n}{j},b)-b \sum_{j=1}^{\lfloor \frac{n}{b} \rfloor}E_{k-1}(\frac{n}{j ...

**7**

votes

**1**answer

405 views

### Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?

For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq ...

**21**

votes

**2**answers

1k views

### What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$

Suppose $m$ is a positive integer. A quantity of interest is
$$
H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right)
$$
The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples ...

**5**

votes

**1**answer

443 views

### Other implications of Zhang's method

I have been reading a bit about Zhang's proof and the associated Polymath8 project.
Though Tao's high level summary
...

**7**

votes

**1**answer

411 views

### The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
...

**8**

votes

**1**answer

333 views

### Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...

**16**

votes

**1**answer

712 views

### Infinitely many primes, and Mobius randomness in sparse sets

Problem 1: Find a (not extremely artificial) set A of integers so that for every $n$, $|A\cap [n]| \le n^{0.499}$, ($[n]=\{1,2,...,n\}$,) where you can prove that $A$ contains infinitely many primes.
...

**8**

votes

**1**answer

503 views

### Density of prime pairs whose gap is less than the average gap

By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...

**1**

vote

**1**answer

149 views

### Behavior of a quantity related to Fermat's 4n + 1 Theorem

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to ...

**3**

votes

**1**answer

348 views

### Heuristic for Montgomery's conjecture

This is my third question on this site regarding Montgomery's conjecture -- and I apologize
if this is too much -- but I am still not understanding well why this conjecture is believed to be true.
...

**61**

votes

**6**answers

6k views

### Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin ...

**16**

votes

**1**answer

4k views

### Tightening Zhang's bound

Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang.
The original bound was ...

**4**

votes

**1**answer

199 views

### Estimate on the prime-counting function $\psi(x)$.

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...

**6**

votes

**6**answers

754 views

### Sequences equidistributed modulo 1

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.
H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.
...

**3**

votes

**1**answer

261 views

### Least non primitive root

There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more generally
an odd prime ...

**4**

votes

**3**answers

264 views

### A divergent series related to the number of divisors of of p-1

Let $d(n)$ denote the number of divisors of $n$. Is it known that the series
$$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$
diverges?
This would follow immediately from the Sophie Germain Conjecture. ...

**1**

vote

**1**answer

164 views

### What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$

I was reading through Jitsuro Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25$.
In the paper, he uses the following bounds for the second Chebyshev function ...

**7**

votes

**0**answers

244 views

### Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...

**27**

votes

**1**answer

863 views

### Prime Number Races in 2 Dimensions

Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto
\sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective?
In 1999, when I was an undergraduate student, ...

**7**

votes

**1**answer

511 views

### Least prime primitive root

For $p$ a prime number, let $G(p)$ be the least prime $q$ such that $q$ is a primitive root mod $p$, that is $q$ generates the multiplicative group $(\mathbb Z/p\mathbb Z$)* .
Is it known that ...

**2**

votes

**1**answer

376 views

### The tightest prime zipper

Define a prime zipper as an increasing function $f(n)$ mapping $\mathbb{N}$ into $\mathbb{N}$
with the property that, for every $n \ge 1$, there is at least one prime within the
inclusive interval $[ ...

**5**

votes

**2**answers

304 views

### Density of integers $n$ whose totient $\varphi(n)$ is larger than $\alpha n$

Fix $0 < \alpha < 1$ a real. Let $S_\alpha$ the set of integers $n \geq 1$ such that be $\phi(n)>\alpha n$. For $x>0$, let $S_\alpha(x)$ be the number of positive integers $n$ less han $x$ ...

**9**

votes

**1**answer

523 views

### Prime Power Gaps

In 2000, Baker, Harman and Pintz proved that there is always a prime in
the interval $(n-n^{0.525}, n)$. There are also conditional results
implying smaller intervals. Nevertheless, I could not find ...

**13**

votes

**1**answer

796 views

### Small primes in arithmetic sequences

Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi_{n,1}(an)$ the number of primes
$p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let
...

**5**

votes

**1**answer

221 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 ...

**4**

votes

**3**answers

583 views

### Better error bounds for partial sums of reciprocals of primes?

One of Mertens' theorems gives that
$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$
where $B$ is a constant near $0.26$ in value and $E(k)$ is an error
term whose size is ...

**6**

votes

**2**answers

503 views

### On a sum involving prime numbers

I find myself needing the asymtotics of the following summation for my work. Let $a$ be a positive real number and $p_n$ be the $n$-th prime.
$$
\sum_{k=1}^{n} [k^a - (k-1)^a]p_k
$$
At $a=1$, this ...

**6**

votes

**1**answer

529 views

### Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that.
$$
p_n = n ...

**4**

votes

**1**answer

451 views

### Progress on Bouniakowsky's Conjecture

Has there been any progress on the Bouniakowsky conjecture? In particular, has anyone been able to prove something for a particular polynomial - or for a class of them?
(I can't seem to find ...

**2**

votes

**1**answer

532 views

### Generalization of Mertens' theorem

One classical Mertens' theorem tells us that $$\prod_{p \leq n} (1-\frac{1}{p})^{-1} = e^\gamma \log n + \mathcal{O}(1).$$
It is now very natural to ask, whether we have some good estimate to ...

**12**

votes

**3**answers

816 views

### Density of a set of integers

EDIT: this question of mine has received little attention, perhaps in part because it was stated in a too general and complicated way. So let me give it a second chance:
Fix an integer $r \geq 0$. ...

**18**

votes

**3**answers

853 views

### Finite sums of prime numbers $\geq x$

Let $S_x$ be the set of finite sums of prime numbers $\geq x$. In other words, let $S_x$ be the submonoid of $(\mathbf{Z}_{\geq 0},+)$ generated by the set $\mathcal{P}_{\geq x}$ of prime numbers ...

**4**

votes

**2**answers

326 views

### Distribution of primes in small intervals

Let $\pi(x)$ be the number of primes smaller than $x$. Do there exist unconditionally universal constants $c > d$ such that
$$
\lim_{x \rightarrow \infty} \frac{\pi(x + \log^c x) - ...

**0**

votes

**2**answers

364 views

### Dirichlet's theorem on prime density [closed]

Does anyone knows where I could find the proof of Dirichlet's theorem on the analytic density of primes congruent to a certain integer $m$, which turns out to be $\frac{1}{\varphi(m)}$?
Thanks a lot.
...

**8**

votes

**1**answer

416 views

### On the least prime in arithmetic progressions

My question concerns the least prime (denoted $p(a, q)$) in the arithmetic progression $a \pmod q$ where $a$ and $q$ are coprime. Quite a time ago Linnik demonstrated that
$$p(a, q) \ll q^L$$
for ...

**-1**

votes

**1**answer

772 views

### 1895 Math Trip problem on primitive roots of unity

How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...

**7**

votes

**2**answers

490 views

### The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$

Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...

**2**

votes

**2**answers

886 views

### Numbers of a different order?

Let $d_r$ be a divergent series of positive terms and let $s_r = \sum_{i=1}^{r}d_r$. We are interested in the sequence of numbers $S_{d_r} = s_1, s_2, \ldots$. For example if $d_r = 1/r$ the $s_r = ...

**5**

votes

**1**answer

416 views

### Analogue of van der Corput sequence for prime numbers

A van der Corput sequence is a low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by placing a decimal point and ...

**6**

votes

**1**answer

446 views

### Logarithmic Integral of n^Zeta Zeroes and certain nested sums of the fractional part function

Below is an approach I've been exploring for connecting the prime counting function with the logarithmic integral and expressing the error term between the two. I find it beguiling, but I've largely ...

**19**

votes

**6**answers

3k views

### explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...

**4**

votes

**2**answers

517 views

### How are these number-theoretical constants actually distributed?

I'm very curious about this and would be really grateful for any help or comments in this direction. If we consider any of the following number-theoretical constants:
1)The various singular series ...

**5**

votes

**2**answers

960 views

### least prime in a arithmetic progression

Hello
Here I want to consider the simplest arithmetic progression $n\equiv 1\pmod{q}$ where $q$ is a prime. Is it true that we can find a prime $p\leq q^2$ in this arithmetic progression?
This ...