All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
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Probability distribution from equidistribution - II
Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$ and denote $N(a,b)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
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0
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116
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Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
CommunityBot
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3
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131
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Chen primes and permutations
In 1973 the Chinese mathematician J.-R. Chen proved that there are infinitely many primes $p$ such that $p+2$ is a product of at most two primes. Nowadays such primes $p$ are called Chen primes.
For $...
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Prime generating polynomials
Continuation to this previous question.
According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...
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Is the following weak version of second Hardy-Littlewood conjecture already known?
Very recently I was going through my previous MSE posts and I stumbled upon some of them regarding the Second Hardy-Littlewood Conjecture which states that,
For all $x,y\ge 2$ we have, $$\pi(x)+\...
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Conjecture: $ x\leqslant f(x)\leqslant x+x^{\log_{113}13} \{x|1\leqslant x\leqslant+\infty,x\in \mathrm{positive~integer}\}$?
Function f(x) is the most closest prime number not less than $x$.
$f(5)=5\qquad f(9)=11$
Conjecture: $ x\leqslant f(x)\leqslant x+x^{\log_{113}13} \{x|1\leqslant x\leqslant+\infty,x\in \mathrm{...
- 105k
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Who first proved that there are at least n^(1-ε) primes up to n?
It's well-known that Hadamard and de la Vallée-Poussin independently proved the Prime Number Theorem in 1896: that $\pi(n)=n/\log n+o(n/\log n)$. I'm curious as to a weaker result: that for any $\...
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consecutive prime gaps and explicit bound
I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" ...
- 105k
3
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Almost-Primes in Short Intervals
Let $S$ be the set of integers which are a product of $k$ distinct primes, $k$ a fixed positive integer (the condition that the primes are distinct is not crucial). Landau used the Prime Number ...
- 4,713
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What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?
What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs ...
- 5,470
11
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1
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934
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Riemann sum formula for definite integral using prime numbers
I had asked this question in MSE. It got lot of upvotes but no answer (except one which was too long to be posted as a comment) hence I am posting it in MO.
While answering another question in MSE I ...
- 4,991
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2
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A stronger form of the Dirichlet Theorem on prime numbers in arithmetic sequences
Question 1. Let $a,b>1$ be two natural numbers. Is there a prime number $p\in 1+b\mathbb N$ such that $a+p\mathbb Z$ is a generator of the multiplicative group of the field $\mathbb Z/p\mathbb Z$?
...
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6
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2
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411
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A simultaneous generalization of the Grunwald-Wang and Dirichlet Theorems on primes
By Grunwald-Wang Theorem, if for some odd number $n$ the equation $x^n=a$ has no solutions in $\mathbb Z$, then the equation $x^n=a\mod p$ has no solutions for some prime number $p$. I am interested ...
- 20.8k
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Are there infinitely many primes of this form?
The semiprime $87 = 3*29$ has a curious property: it's the fact that both
$87^2 + 29^2 + 3^2 = 8419$
and
$87^2 - 29^2 - 3^2 = 6719$
are prime numbers.
This intrigued me and led me to wonder if ...
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Goldbach's conjecture for the Liouville function
Is it true that for every even integer $N > 2$, there exist positive integers $a,b$ such that $a + b = N$ and $\lambda(a) = \lambda(b) = -1$ ?
Here $\lambda$ is the Liouville function.
10
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Spacing of fractions with prime denominator
Let $X\ge 2$ be large. Let $$A = \left\{\frac{a}{q}: 1\le a < q\le X,\ (a, q) = 1\right\},$$ and let $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ ...
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Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?
I am looking for a comment, reference, remark, or proof of three conjectures as follows:
Conjecture 1: Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P_{n+...
CommunityBot
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On the regularity of integer solutions of a simultaneous equation with consecutive prime coefficients
Let $p_1$ through $p_6$ be consecutive primes in ascending order, and consider the simultaneous equation $$p_1x+p_2y=p_3\\p_4x+p_5y=p_6$$
Motivating Question: For what $p_1$ does the system provide ...
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2
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$\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?
There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples:
Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ ...
- 43.7k
4
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1
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414
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Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $?
Disclaimer : the following reasoning is a physicist's one and as such may not be suitable for this website. Still it may give rise to potentially interesting insights so I ask it anyway.
I stumbled ...
- 1,951
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206
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Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
- 12.8k
1
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1
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Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?
I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:
For $n, x \ge 2$ be two integers then:
$$P_{2n} \ge 2P_n$$
and
$$\pi(2x) \le 2\pi(x)...
- 105k
-3
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1
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245
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Can this weakening of Polignac's conjecture be proven?
Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the ...
- 15.6k
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1
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154
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Sergei numbers : even integers n being a prime gap at least n times
Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number ...
CommunityBot
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What is known about stability of number theoretic statements for Beurling systems which are based on small perturbations of the ordinary primes
Beurling considered a sequence of reals $1<x_1<x_2<\cdots <$ as "primes" and then the ordered sequence of all products of these "primes" as "integers". Let us consider Beurling primes ...
- 24.7k
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411
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Sum of reciprocals of integers minus primes
For any integer $m>2$, let $P_m$ be the set of primes less than $m$, and let
$$
f(m) = \sum\limits_{p \in P_m} \frac{1}{m-p}.
$$
For example, $f(3)=\frac{1}{3-2}=1$, $f(4)=\frac{1}{4-2}+\frac{1}{4-...
- 105k
3
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0
answers
125
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Number of prime differences
Has any progress been made since Chen on bounding
\begin{equation*}
G(n) = \#\{\epsilon N < p_1, p_2 \leq N: n = p_1 - p_2\}
\end{equation*}
from above?
As far as I can tell, the best upper ...
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probability of m to be a primality radius of n
Disclaimer : this is a crosspost from MSE, as the question got one upvote but no comment or answer whatsoever.
Assuming Goldbach's conjecture, let's denote by $r_{ 0}(n):=\inf\{r\geq 0,(n−r,n+r)\in\...
CommunityBot
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Use of weights in the GPY's and Tao-Maynard's work on the twin prime conjecture
I am going through James Maynard's paper, Small Gaps between Primes, and have a number of questions regarding his approach. First, I am wondering why uses weights in his approach. While I generally ...
- 105k
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142
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Mobius function on values of an irreducible quadratic polynomial
Are there infinitely many integers $n$ for which $n^2 + 1$ is square-free, and has an even number of (necessarily distinct) prime factors ?
- 11.3k
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Compare $\operatorname{rad}(an+b)$ and $\varphi(cn+d)$ in a simple and interesting inequality, for some choice of integers $a,b,c$ and $d$
We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this ...
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96
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Are the elements in the n-th row of the first matrix a permutation of the elements in the n-th row of the second matrix?
From my previous questions here and here the following two matrices arise for twin primes and cousin primes from Dirichlet convolution:
For $h=2$ twin primes:
$$T_2(n,m)=\sum\limits_{\substack{k=1 \\...
- 1,183
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1
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Moments of merit
The merit of a prime gap equals $(p_{n+1}-p_n)/\ln p_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very ...
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13
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2
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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.
$$
p_n = n \...
- 655
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Deduction of the classical Halasz's inequality from the new form
In the new proof of Halasz's Theorem, the authors give a statement in terms of a quantity $L(x)$ defined by $$L(x)^2=\sum_{|N|\leq\log^2x+1}\frac1{N^2+1}\max_{|t-N|\leq1/2}|F_x(1+it)|^2,$$
where $F_x(...
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1
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Asymptotic for a number theoretic sequence and its Dirichlet series' convergence
I would like to know the asymptotic behaviour at large $n$ for $t\in\mathbb{R}$, $t\neq0$ of the following function:
\begin{align*}
A_n(t)&=\sum_{q=\frac{a}{b}\in \mathbb{Q}^+|\gcd(a,b)=1 \& ...
CommunityBot
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2
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Distribution of primes in small intervals
Let $\pi(x)$ be the number of primes smaller than $x$. Do there exist unconditionally universal constants $c > d$ such that
$$
\lim_{x \rightarrow \infty} \frac{\pi(x + \log^c x) - \pi(x)}{\log^{c-...
- 6,984
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1
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350
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Counting smooth numbers in short intervals
I am reading a few papers about counting smooth numbers in the interval $[x, x+\sqrt{x}]$, including the work of Harman, and Matomaki.
Both authors mentioned that the Dirichlet polynomial techniques ...
- 36
3
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1
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328
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Estimating a sum over prime numbers
At page 3 of this article https://arxiv.org/pdf/1706.03755.pdf about the new proof of Halasz's Theorem, the authors claim that $$\sum\limits_{(\log x)^4< p\leq x/2}\frac{\log p}{p\log(x/p)}\ll\log\...
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Twin prime based Dirichlet series
Assuming there are infinitely many twin primes, one can consider a Dirichlet series $ \sum_{n>0}a_{n}{n^{-s}} $ and replace the sequence of positive integers with the sequence of twin primes. That ...
- 105k
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319
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Isometry group of an integer
This is a cross post from MSE, as it seems the partial answer I got then was deleted, so I ask it again here.
Let $\prod_{i\in I}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ ...
- 6,984
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Primes of the form $x^2 + y^2 + 1$
There are infinitely many primes of the form $x^2+y^2+1$, as proved by Bredihin. Motohashi improved the result by showing that there were $\gg x/\log^2 x$ such primes up to $x$. But we expect $\Theta(...
- 43.7k
3
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153
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Short intervals containing a prescribed number of primes
In arXiv:1802.10327, the following conjecture appears:
$$\vert\{n\leq x,\vert [ n,n+\lambda\log n]\cap\mathbb{P}\vert=m\}\vert\sim\frac{\lambda^{m}e^{-\lambda}}{m ! }\cdot x$$
Differentiating the ...
- 6,406
1
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1
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130
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Prime inequality $P_{n.m}$ $<$ $P^{m}_n$ [closed]
Let $P_n$ be nth prime, where $n,m \in$ $\mathbb{N}$, $n,m >1$.
How i can show that $P_{n.m}$ $<$ $P^{m}_n$.
I tried to used $PNT$ but that feels like overkill, is this inequality already ...
- 105k
0
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1
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198
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Upper bound for $\sum r_{0}(n)$
Assuming Goldbach's conjecture, let's define for a sufficiently large integer $ n $ the quantity $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $.
Under GRH, what is the best upper bound ...
- 105k
1
vote
0
answers
141
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On certain number theoretic sextuples?
Given small parameters $0<\epsilon<\epsilon'$ is there an $n_\epsilon>0$ such that at every $n>n_\epsilon$ if we are given a prime $n^2<p<2n^2$ then can we always find integers $a,b,...
- 13.9k
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1
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834
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Is my proof a notable result? If so, where and how do I publish it? [closed]
I can prove that given $ε$ chosen arbitrarily small, if $\prod_{p \le p_k} p^{\frac{1}{p-1}} \lt \frac{1 + ε}{e} p_k$ then $∀n\ge p_k∃p∈\mathbb{P} | n \le p \lt (1 + ε)n$.
Actually this result is ...
- 105k
53
votes
5
answers
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Distribution of square roots mod 1
I was wondering about the distribution of $\sqrt{p}$ mod $1$ this morning, as one does while brushing one's teeth. I remembered the paper of Elkies and McMullen (Duke Math. J. 123 (2004), no. 1, 95–...
- 2,207
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1
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120
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How many integers $x$ satisfy that $x*p(x) \leq n$, where $p(x)$ means the largest prime factor of $x$?
I guess that the number of integers $x$ which satisfy the condition $x*p(x) \leq n$ is $O(n^{2/3})$ or $O(n^{3/4} / \ln n$), but I cannot prove it. I just write a program to count the number. The ...
- 109k
6
votes
1
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246
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The L-function of Q(-1/2) and the "number of prime $p\equiv 3$ divisors" function
In the framework of classical motives, there is no such thing as a motive $\mathbb Q(-\tfrac 12)$, i.e. a tensor root of $\mathbb Q(-1)$. There is one, however, in a more general setting of "...
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