**1**

vote

**0**answers

118 views

### Identity on sum over reciprocals of prime products?

The following identity seems to follow from a simple analysis of the sieve of Eratosthenes and inclusion-exclusion, where $p_i, p_j, p_k, \ldots$ denote primes and $N$ is an integer $\geq 2$:
...

**4**

votes

**2**answers

200 views

### Orders of the conjugates of an algebraic prime number in its residue field

Of interest to me is the following question (it would be nice to find out what is known in its direction):
Given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime ...

**1**

vote

**0**answers

68 views

### Error analysis needed for more refined estimates (than Salat-Zanam) of the sum of prime powers

A recent question on math overflow on sums of the primes squared was answered/put on hold by pointing to an old paper by T. Salát and S. Znám, On the sums of the prime powers. Salat and Znam's (SZ) ...

**1**

vote

**1**answer

174 views

### A square-squareroot integer race sequence involving primes

I wonder what is the expected behavior of this process?
Let
$f^2_{\mathrm{next}}(n) =$ the next prime after $n^2$.
$g_{\mathrm{sqrt}}(n) = \lfloor \sqrt{n} \rfloor$.
Now iterate as ...

**3**

votes

**1**answer

165 views

### Wiener-Ikehara tauberian theorem and order of pole at s=1

In the introduction to Akshay Venkatesh's thesis "Limiting Forms of the Trace Formula" we have the following statement :
"For, in summing over primes, the limit
...

**2**

votes

**1**answer

156 views

### About a Variant of Ulam Spiral

I asked this on Math.SE but got no answer:
Here I read about a variant on the Ulam spiral:
[A] structure may be seen when composite numbers are also included in
the Ulam spiral. [...] Using ...

**2**

votes

**1**answer

207 views

### Number theoretic functions that have an irregular behaviour at primes

Usually, number theoretic functions have "trivial" (or at least easily defined) values for primes. In this thread, I am rather asking for functions which are only defined on primes (well, this ...

**7**

votes

**1**answer

351 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 ...

**2**

votes

**1**answer

292 views

### Is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture?

The question is in the title: is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture? I tried to give a heuristics in Upper bound for $r_{0}(n)$ through probabilities that seems ...

**0**

votes

**0**answers

123 views

### Does Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?

Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...

**5**

votes

**1**answer

486 views

### Has this strengthening of the PNT already been conjectured?

Suppose $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ is an arithmetic function that grows slower than the identity map. Has it already been conjectured that, under this general hypotheses, ...

**35**

votes

**1**answer

3k views

### Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset logarithmic ...

**2**

votes

**1**answer

435 views

### On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski

In response to a comment posted under
Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd ...

**1**

vote

**3**answers

228 views

### Powers of $2$ and the products of initial odd primes

NOTATION: $O_x$ -- the product of all odd primes $\le x$.
E.g. $O_7=3\cdot 5\cdot 7 = 105$.
QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the ...

**11**

votes

**1**answer

401 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 ...

**0**

votes

**2**answers

504 views

### Yitang Zhang's paper [closed]

I just want make thing clear for myself. Others may have asked before in different ways. Does Yitang Zhang's paper prove that for any given length gap $g_n > N$ there is a prime $p_n$ for which ...

**2**

votes

**1**answer

333 views

### Looking for paper: Weil's original 1952 “Sur les formules explicites de la théorie des nombres premiers”

I am looking for a source (preferably online) for Weil's original 1952 paper on the explicit formula. I am aware of an english translation available here, but would like to have access to the original ...

**3**

votes

**1**answer

163 views

### Congruences among primes modulo which a given polynomial has roots

Suppose $f(x)\in\mathbf Z[x]$ is nonconstant. I would like to know if either of the following statements is true.
If $a$ and $b$ are coprime integers (probably with some additional restriction), ...

**7**

votes

**0**answers

239 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 ...

**4**

votes

**0**answers

159 views

### Can the following quantitative version of Chen's theorem be obtained?

The theorem of the Chinese mathematician Chen Jingrun is currently one of the best results with respect to the binary Goldbach problem. It asserts that every even integer $n$ is in the sumset $P + ...

**5**

votes

**0**answers

240 views

### Primes for which 2 is a primitive root

I am writing a paper in which I keep referring to primes p for which 2 is a primitive root mod p and so I want to give a name for these primes. Is there a name for these primes in the literature ...

**1**

vote

**1**answer

158 views

### First Parameterized Subset of Primes that was Related to a Mathematical Result

To my knowledge, Fermat primes, i.e. primes of the form $2^{2^n}+1$ were the first to play a role in a mathematical result, namely in the characterization of constructible regular n-gons. Gauss ...

**6**

votes

**1**answer

571 views

### What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...

**6**

votes

**0**answers

301 views

### Twin Primes that are Sophie Germain Primes

Suppose $p$ is a prime such that $p + 2$ is also prime, and nothing else is known about $p$.
Is there any reason to think that this affects the probability that $p$ is also a Sophie Germain prime? ...

**3**

votes

**1**answer

135 views

### Decidability of prime gap sequences

Is the following problem undecidable?
Given a sequence of $n$ gaps $d_1,d_2,...,d_n$, does there exist a sequence of $n+1$ primes $p_1,p_2,...,p_{n+1}$ such that $p_{i+1} - p_i = d_i$ ?
If not, is ...

**0**

votes

**0**answers

150 views

### Average order and upper bound of $r_{0}(n)$

Assume Goldbach's conjecture. Then for every integer $n>1$ there exists a non-negative integer $r$ such that $n-r$ and $n+r$ are both primes. For a given $n>1$, the smallest such $r$ will be ...

**26**

votes

**4**answers

3k views

### What is exceptional about the prime numbers 2 and 3?

Admittedly this question is vague. But I hope to convey my point. Feel free to downvote this.
Permit me to define prime number the following way:
A number $n>1$ is a prime if all integers $d$ ...

**6**

votes

**1**answer

214 views

### Question about a certain class of primes

I've come across a set of primes in a problem I'm working on, and I'm wondering if there's more information available about them. I'm guessing not much, particularly since the question of infinitude ...

**1**

vote

**0**answers

197 views

### An inequality about Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...

**1**

vote

**2**answers

243 views

### Infinite play with tape, or covering the integers with prime arithmetic progressions

It is possible that a more technical version of this question has been
asked and answered in the literature. If so, then a reference is much
appreciated. I will phrase it in terms of colored tapes ...

**2**

votes

**1**answer

81 views

### Principally split primes with factors in arbitrarily small angular sectors

I wonder if the following is known:
let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where
$x, y$ ...

**3**

votes

**0**answers

129 views

### An estimate for dividing n^2 by each of the primes up to and including n, and then summing the results [closed]

I know that the asymptotic for the sum of all the primes up to n is $n^2/2\log n$. But I'm trying to find the formula (an estimate) for when $n^2$ is divided by each of the primes up to $n$, in turn ...

**3**

votes

**1**answer

440 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 ...

**5**

votes

**2**answers

280 views

### Sum of digits of repeating end of reciprocal of prime over period is $\frac{9}{2}$

Take a prime other than 2,3 or 5 and look at the part of it that repeats in base 10. Is it true that the sum of the digits in the end divided by the period(number of repeated digits id always ...

**2**

votes

**3**answers

371 views

### Does this 'alternating' Euler product converge for all $\Re(s) > 0$?

Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ?
$$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} ...

**0**

votes

**3**answers

570 views

### Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...

**2**

votes

**2**answers

234 views

### What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$?

I know the following:
Proven: There is a prime number between $n$ and $2n$ for every integer $n>0$.
Conjectured: There is a prime number between $n^2$ and $(n+1)^2$ for every integer $n>0$.
...

**1**

vote

**2**answers

253 views

### prime zeta function when $0<s<1$ [closed]

I will not be surprised if this question seems trivial in MO but i asked it first in MathSE and i did not get an answer.
So, here it is:
I would like to know if there is a good estimate for the sum ...

**5**

votes

**1**answer

455 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 ...

**3**

votes

**1**answer

323 views

### Giuga's Conjecture: Central or Peripheral?

An earlier MO question
highlighted
Giuga's Conjecture:
A positive integer $n>1$ is prime if and only if
$$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$
For example, for the prime $n=5$, ...

**3**

votes

**0**answers

183 views

### Logarithms of ratios of squarefree numbers

Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes.
For example, for $M=2$ and $N=4$ you get ...

**11**

votes

**1**answer

512 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

**1**answer

367 views

### Does this prime-gaps pattern occur infinitely often?

Let $p_n$ be the $n$-th prime.
For each integer $k \ge 0$, do there exist
an infinite number of $k+3$ consecutive primes
$(p_n, p_{n+1}, \ldots, p_{n+2+k})$
so that
(1) The gap between the 1st and ...

**-2**

votes

**1**answer

197 views

### Giuga and Carmichael numbers

If $p$ is both Giuga and Carmichael number
then its known that
$1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1} \equiv -1\pmod{p}$
is it true that
if $p$ is both Giuga and Carmichael number then
...

**0**

votes

**0**answers

161 views

### Interval containing prime numbers

Let $\varepsilon$ be an arbitrary small positive number. Can we prove that there exist an $n\in Z$ such that the interval $[2^n,(1+\varepsilon)2^n]$ contain a prime number?

**5**

votes

**0**answers

244 views

### Should I expect to see numbers this smooth?

I have a sequence $N_k$ of numbers whose growth I wish to determine, or at least
approximate nicely. When I look at the ratios of consecutive members,
I find some interesting simplifications ...

**-1**

votes

**1**answer

113 views

### Fermat pseudo prime base-3 [closed]

Good morning!
I have checked the following statement by random numbers of my choice. I am seriously looking for proof of the statement.
Statement: $m$ is said to be Fermat pseudo prime in base-3, ...

**2**

votes

**0**answers

123 views

### Analytic varieties for the primes and the twin primes

I am wondering what real and complex analysis say
about the primes and twin primes.
According to Wikipedia
analytic variety is defined locally as the set of common zeros of finitely many analytic ...

**5**

votes

**3**answers

322 views

### Weak versions of Bertrand's postulate

We are interested in the following statement:
For each $n>1$ and $x>2$ there is at least one prime $p$ satisfying $x<p<n x$.
For $n=2$ we get precisely the Bertrand's postulate which is ...

**0**

votes

**0**answers

67 views

### existence of elements with specific norms in pure cubic fields

Is there any specific way to find an element with a given norm in pure cubic field? say for an example an element of norm 5 in pure (monogenic) cubic field of 11.
it is easy to check that 5 as an ...