7
votes
1answer
261 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 ...
1
vote
1answer
201 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
0answers
76 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
1answer
435 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, ...
-4
votes
0answers
79 views

Where to include contact details in math paper? [closed]

I recently submitted a paper to a math journal on a prime number patter using latex formatting, but they sent an email back saying that the contact details for the corresponding author should be in ...
27
votes
1answer
2k 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
1answer
391 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
3answers
212 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 ...
9
votes
1answer
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 ...
0
votes
2answers
408 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
1answer
242 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
1answer
137 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), ...
3
votes
0answers
136 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
0answers
209 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 ...
4
votes
1answer
432 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
0answers
245 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? ...
1
vote
0answers
110 views

Lehmer's Totient Problem

Recently while researching on the famous Lehmer's Totient Problem I have found that only counterexamples can arise from the following conditions- Let $n$ be a odd positive Carmichael Number and let ...
0
votes
0answers
143 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 ...
6
votes
1answer
200 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
0answers
176 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
2answers
218 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
1answer
70 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
0answers
127 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
1answer
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 ...
5
votes
2answers
219 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
3answers
292 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
3answers
527 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
2answers
208 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
2answers
217 views

prime zeta function when $0<s<1$

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 ...
4
votes
1answer
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 ...
3
votes
1answer
304 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
0answers
173 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 ...
10
votes
1answer
401 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
1answer
309 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
1answer
179 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
0answers
141 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
0answers
227 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
1answer
97 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
0answers
116 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
3answers
302 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 ...
4
votes
2answers
284 views

Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture Let $f(x)$ be non-constant irreducible polynomial with integer coefficients, no fixed prime factor and positive leading coefficient. Let $S$ be a finite set of ...
1
vote
0answers
115 views

Updated tables of maximal prime gaps? [closed]

The website http://www.trnicely.net/gaps/gaplist.html contains a long list of maximal and nonmaximal prime gaps. In this list, the largest maximal prime gap is one of length 1476: Size: 1476 Gap ...
7
votes
1answer
293 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 ...
1
vote
0answers
80 views

For which types of problems can one expect to use Bombieri-Vinogradov in place of GRH?

I must profess a general ignorance of problems that were once known to be true under GRH but has since became unconditional due to the Bombieri-Vinogradov theorem, but I am aware of the heuristic that ...
4
votes
1answer
138 views

Higher dimensional generalization of the Hardy-Littlewood conjecture?

The famous Hardy-Littlewood conjecture on prime-tuples states that if $\{h_1, \cdots, h_k\} = \mathcal{H}$ is an admissible set, that is, for every prime $p$ the set $\mathcal{H}$ does not contain a ...
15
votes
1answer
657 views

Sums of primes that are themselves prime

I'm not a math expert so this may be a trivial question; if $p_i$ is the $i$-th prime, let: $$S(n) = \sum_{i=1}^n p_i$$ be the sum of the first $n$ primes and $$P(n) = | \{1 \leq i \leq n \mid ...
0
votes
1answer
120 views

Tail of singular series of Goldbach problem

Let $N$ a large number and $P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$ ...
12
votes
2answers
852 views

Can I express any odd number with a power of two minus a prime?

I have been running a computer program trying to see if I can represent any odd number in the form of $$2^a - b$$ With b as a prime number. I have seen an earlier proof about Cohen and Selfridge ...
4
votes
0answers
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 ...
17
votes
2answers
707 views

Floors of rationals to powers: Infinite number of primes?

Let $r=a/b$ be a rational number in lowest terms, larger than $1$, and not an integer (so $b > 1$). Q. Does the sequence $$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor, ...