All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
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About Omega prime function
Let $ω(n)$ be the number of distinct prime factors of $n$.
Is the inequality $ω(n)\leq C\log\log(n)$ true and if so what is the value of the constant $C$ ?
- 31
0
votes
0
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91
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How to use prime number theorem In such cases?
Let,
$$A(x)=\sum_{p\leq x}f(p)$$
Where $p$ is a prime number.
Under the Prime Number theorem we have that,
$$\pi(x)=Li(x)+O\left(\frac{x}{e^{a\sqrt{\ln(x)}}}\right) $$
as $x$ approach infinity.
Now,
$$...
- 111
0
votes
0
answers
80
views
Relevance of the deduction of similar theorems than Maier's theorem for other prime constellations
A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...
3
votes
1
answer
348
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On conjectures about the arithmetic function that counts the number of Sophie Germain primes
I've edited this post two years ago on Mathematics Stack Exchange, with identifier 3590406 and same title On conjectures about the arithmetic function that counts the number of Sophie Germain primes, ...
1
vote
0
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482
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Explicit formula for zeta function with special type of weight
Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \...
- 782
2
votes
0
answers
244
views
Lower bounding the number of Galois radii of an integer
Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.
Does it suffice to ...
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11
votes
2
answers
1k
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Do consecutive integers have a big prime factor?
Let us say that three consecutive positive integers $(m-1,m,m+1)$ have a big prime factor if the largest prime factor $p$ of $N=(m-1)m(m+1)$ satisfies $e^p>N$.
I ckecked that it is true for all $m&...
0
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0
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89
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A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel
I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
-1
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1
answer
250
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Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$
Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving ...
user485483
2
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0
answers
263
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Selberg's 1943 result on primes in short intervals and primality radius
This preprint: https://arxiv.org/abs/2207.05038 states in the last paragraph of the first page that a result of Selberg (1943) implies that under RH, almost all intervals of the form $(x,x+\left(\log ...
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9
votes
1
answer
422
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Getting prime by changing 2 digits
I just got a result, an exercise of Tenenbaum's book that one cannot get a prime from arbitrary natural number $n$ by changing only one digit of its decimal expansion. For example, you cannot get ...
- 261
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1
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611
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Why does this convolution of the prime counting function $\pi$ look like a parabola?
In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture:
$$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$
The ...
- 3,064
2
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0
answers
158
views
Quadratic patterns in summands of Goldbach's conjecture
Let $n $ be even and define
$$ Q(n)=\sum_{\substack{ p,q \ \textrm{ primes} \\p+q=n }}\left(\frac{p}{q} \right),$$ where $\left(\frac{p}{q} \right)$ is the quadratic Legendre symbol.
Has this sum been ...
- 3,062
2
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0
answers
125
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Conditional stronger bounds on Linnik theorem with prime power modulus
This post is related to questions asked here and here. However, I include the relevant background on least prime in arithmetic progressions presented here for benefit of the reader.
By Linnik's ...
- 1,042
5
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1
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392
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Divergence of primes dividing polynomials
Let $Q : \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial. Form the set
$$M_{Q} := \{p:\text{ }p\text{ is prime, }\exists n_{p}\in \mathbb{Z}\text{ so that }p|Q(n_{p})\}$$
Is $$\sum_{s \in M_{Q}}\...
- 227
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0
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83
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Prime powers gap of type $(a,b)$
For $n$ a given positive integer, say $r$ is a Galois radius of $n$ of type $(a,b)$, level $l=ab$ and rank $\rho=a+b$ if $n-r=p^a$ and $n+r=q^b$ with both $p$ and $q$ prime.
Denote by $PPG_{a,b}(m)$ ...
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2
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1
answer
159
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Is the density of integers $n$ such that the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ is surjective positive?
Let $\omega(m)$ be the number of prime factors of $m$ regardless of multiplicity. I'm interested in the behavior of the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ for a given integer ...
- 6,974
2
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1
answer
545
views
Is there a Cramer's conjecture for Sophie Germain primes?
A prime $q$ such that $2q+1$ is also a prime is a Sophie Germain prime.
Cramer's conjecture tells gap between consecutive primes is bound by $O(\log^2p)$.
Is there a similar conjecture for Sophie ...
- 13.9k
9
votes
1
answer
400
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The difference between consecutive primes in arithmetic progressions
Let $\pi(x)=\sum_{p\leq x}$ denote the prime counting function. A well known result of Baker, Harman, and Pintz on prime gaps states that for $x\geq y\geq x^{0.525}$ we have that
$$\pi(x+y)-\pi(x)\gg \...
- 11.4k
3
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0
answers
158
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What can be said about the primality of Zsigmondy numbers?
I am cross-posting this from math.stackexchange, as it has received upvotes but no comments/answers after a couple months.
Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ be the $n$-...
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2
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A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis
Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...
- 1,018
3
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0
answers
428
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Proof of an explicit formula for $\pi_0(x)$
Let $\pi(x)$ denote the prime counting function and $$\pi_0(x) = \lim_{\epsilon \to 0} \frac{\pi(x+\epsilon)+\pi(x-\epsilon)}{2}.$$
I've seen noted in a few references the explicit formula
$$\pi_0(x) =...
- 4,985
8
votes
2
answers
814
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Estimates about prime numbers: a lemma in Bourgain's article
For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds:
\begin{equation}
|\{q\in \mathbb{N},q<Q: \text{all ...
- 351
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0
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240
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Liu's new sieve weight
Does Liu's sieve weight (in his arXiv paper "On the gap between primes")
$$sieve(n)=(\sum_{\substack{d_i\mid (n-h_i),i=1,\cdots,k\\ (d_1,\cdots,d_k)\in\mathcal{D}}}\lambda_{d_1,\cdots,d_k} ...
- 11
1
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0
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749
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Can the prime gap record of Liu be improved further?
Let $d$ be the least positive integer such that there are infinitely many distinct prime pairs $\{p,q\}$ with $|q-p|\le d$. The twin prime conjecture is equivalent to $d=2$. In 2013 Yitang Zhang ...
- 15.6k
2
votes
0
answers
144
views
A conjecture about prime test
Conjecture If $\varphi(m)<\varphi(n)$ for all $m<n$,then $n$ is a prime number.
I tried to find a counterexample when $n=pq$ ($p,q$ are prime), then we have to find a prime between $(p-1)(q-1)$ ...
- 141
4
votes
1
answer
219
views
Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$
Note: Posting in MO since it was unanswered in MSE
Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
- 5,470
4
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0
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210
views
No perfect patterns in the primes
The primes are equidistributed in the residue classes $1(\!\!\!\mod{4})$ and $3(\!\!\!\mod 4)$. We also know (for example, by Rubinstein-Sarnak) that the patterns cannot be eventually alternating, i.e....
- 3,062
9
votes
3
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584
views
Why is there an unexpected increase in the density of certain types of Goldbach primes?
Note: Posted in MO since it was unanswered in MSE.
I was checking how quickly we can verify Goldbach's conjecture for a given even number $n$ and it was clear that searching backward starting from the ...
- 5,470
7
votes
0
answers
335
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Residues of consecutive primes modulo a fixed integer
It is well-known that the primes are uniformly distributed in residue classes modulo any fixed integer. More precisely, for each integer $q$ and each residue $a \in \mathbb{Z}/q\mathbb{Z}$ that is ...
- 1,914
4
votes
1
answer
235
views
Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number
Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it ...
- 5,470
1
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0
answers
84
views
How common are semiprimes with equally bitsized factors among semiprimes with equal bitsize?
I am curious about the following after having looked at the paper "Almost primes in almost all short intervals", theorem 3 says:
Almost all intervals $[x, x + \log^{3.51}{(x)}]$ with $x ≤ X$...
- 11
5
votes
1
answer
214
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Remainder terms of congruence sums in sets of positive density
Let $\mathcal{A} \subset \mathbb{N}$ be an infinite sequence with positive density, in the sense that
$$
\tag{1}
\lim_{x\to\infty} \frac{|\mathcal{A} \cap x|}{x} = c > 0,
$$
and define the ...
- 1,990
1
vote
1
answer
306
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Expected number of primes of particular size and from a linear form
Given two distinct primes $P_1$ and $P_2$ picked randomly and uniformly in the interval $[T^2,2T^2]$ consider the set $\chi(P_1,P_2)$ of numbers of form $$xP_1-yP_2$$ where $x,y$ are in $[0,T^{1+\...
- 13.9k
9
votes
2
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547
views
Primes between $x$ and $x+x^\theta$
Iwaniec [1] proved that
$$
\pi(x+x^\theta)-\pi(x) < \frac{(2+\varepsilon)x^\theta}{\eta(\theta)\log x},\ x>x_0(\varepsilon,\theta).
$$
with
$$
\eta(\theta)=\frac{15\theta-2}{9}.
$$
(Actually, he ...
- 9,114
10
votes
0
answers
350
views
Are there are any attempts utilising sieve theory to attack the general $a p \pm 1$ problem?
It is currently an open question if there are infinitely many primes $p$ such that $2p + 1$ is prime (Sophie Germain primes) or that at least one of $24p \pm 1$ is prime.
Could Zhang's method, or the ...
- 533
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votes
0
answers
292
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A prime generating algorithm
I posted this question in MSE around a month ago, but didn't receive any suitable answers. So, I decided to give it a try here as well-
I was trying to explain the famous proof of infinitude of primes ...
- 791
1
vote
1
answer
124
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why is $R := \mathbb{Z}p +XF[[X]]$ an almost valuation domain?
Recall that an integral domain $R$ with quotient field $K$ is
an almost valuation domain if for every $0 \not= x \in K$, there is a positive integer
$n$ (depending on $x$) such that $x^n \in R$ or $x^{...
- 147
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votes
1
answer
97
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$l$-th power radioprimal conjecture
I would like to know if some widely believed conjecture, be it GRH, Hardy-Littlewood conjecture, or any other would imply the following statement for some $l>1$:
$l$-th power radioprimal growth ...
- 6,974
10
votes
2
answers
282
views
Non-trivial upper bound for a sum related to $p^{-1}z \pmod q$ and $q^{-1}z \pmod p$
Let $p, q$ be two distinct prime number. I'm trying to provide a non-trivial upper bound for the sum
$$S(p, q) = \sum_{1 \leq x < p} \sum_{1 \leq y < q} \frac{1}{\|x / p\| \, \|y / q\| \, \|x/p +...
- 65
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votes
0
answers
177
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Is there a link between Elliott-Halberstam and weak Hardy-Littlewood-Goldbach conjectures?
Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-...
- 6,974
3
votes
2
answers
465
views
Least number coprime to a given integer
For a positive integer $n$ let $$f(n):=\min\{m\in \mathbb N: m>1, \gcd(m,n)=1\} .$$
Equivalently, $f(n) $ is the smallest prime not dividing $n$.
Is there any upper bound literature for this? It is ...
- 3,062
1
vote
0
answers
293
views
Can a lower bound for this weakening of Goldbach's conjecture be reached?
Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime, and that a non negative integer $w$ is a Galois radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, where $\...
- 6,974
13
votes
2
answers
1k
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What is known about the prime number theorem for Beurling generalised primes
Background: Beurling's systems of numbers
Beurling considered a sequence of reals $1<x_1<x_2<\cdots <$ as "primes" and then the ordered sequence of all products of these "...
- 24.7k
3
votes
0
answers
232
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Numbers made up of primes from a given set
Take a set $\mathcal P$ of primes and denote by $\langle \mathcal P\rangle $ the set of all natural numbers composed of primes from $\mathcal P$. If
\[ \sum _{p\in \mathcal P}\frac {1}{p}\]
converges ...
- 1,381
3
votes
0
answers
117
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On the Carmichael Lambda function
Let Carmichael function be denoted by $\lambda(n)$.
Consider the set $I_m=\{n:\lambda(n)=m\}$.
What is known about the cardinality of $I_m$?
Let $P_m=\{p\in Primes: p|\ell \mbox{ for some }\ell\in ...
- 13.9k
0
votes
0
answers
102
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On simple examples of unimodularity
$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close.
Is there elementary example where only $w$ is even and all four ...
- 13.9k
4
votes
1
answer
400
views
Bounding integrals involving $\operatorname{li}(x)-\pi(x)$
Let $x >0$. How can one find good $O$ bounds on the integrals
$$\int_0^x\frac{\operatorname{li}(t)-\pi(t)}{t}dt$$
and
$$\int_x^\infty\frac{\operatorname{li}(t)-\pi(t)}{t^2}dt$$
where $\pi(x)$ is ...
- 4,985
2
votes
1
answer
461
views
How essential is the vanishing of the Dirichlet $L$-functions to Dirichlet's theorem on primes in arithmetic progressions?
I seem to recall that the prime number theorem (PNT) is equivalent to the fact that the Riemann zeta function $\zeta(s)$ is non-zero on all of $\text{Re}(s) = 1$ (see https://math.stackexchange.com/...
- 831
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0
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326
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Counting primes, twin primes, cousin primes: unusual approach, connection to some conjectures
I am investigating the following sieve-like algorithm. Let $S_N=\{1,\dots,N\}$. For all primes $p$ with $p_0\leq p \leq M$, we remove from $S_N$ the following elements: all numbers $n\in S_N$ such ...
- 3,098