All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
6
votes
1
answer
826
views
Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$
I was recent reading through Paul Erdos's classic elementary proof of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p_i$ represents the $...
- 13k
2
votes
1
answer
188
views
Integers whose product is a primorial and primality of their sum or difference
Let $ a $ and $ b $ be two positive integers such that $ a\lt b $ and $ ab $ is a primorial. Let $\mathcal{N}(x)=\mathcal{N}_{prime}(x)+\mathcal{N}_{pure}(x)+\mathcal{N}_{mixed}(x)$ where $ \...
- 30.1k
22
votes
1
answer
2k
views
Reasons behind assuming the existence of Siegel zeros can be used to prove something stronger than assuming GRH?
There are few results that I am aware of where one can prove something stronger by assuming the existence of Siegel zeros than by assuming the GRH. For example Heath-Brown proved the existence of ...
- 39.9k
2
votes
0
answers
147
views
Skewes' number and the ratio $\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$
(A complementary post is here.)
Given the prime counting function $\pi(x)$ and the logarithmic integral $\operatorname{li}(x)$, we have Table 1,
$$\begin{array}{|c|l|}
\hline
x&\operatorname{li}...
- 12.6k
7
votes
0
answers
179
views
When does the function $F(x)=\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$ reach $F(x) > 8$?
We know from Ramanujan and Riemann that,
$$\pi(x) = \operatorname{li}(x) -\tfrac12\operatorname{li}(x^{1/2})-\tfrac13\operatorname{li}(x^{1/3})-\tfrac15\operatorname{li}(x^{1/5}) +\dots$$
with prime ...
- 12.6k
13
votes
1
answer
899
views
Parity of the Prime Counting Function
I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers.
Let:
$\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 ...
- 63.9k
1
vote
1
answer
356
views
Some questions about some examples in "sieve methods" in the book "Opera de Cribro" by Friedlander and Iwaniec
I am reading the book "Opera de Cribro - John B. Friedlander, Henryk Iwaniec" and in pages 5,6 I do not understand why and how they chose $X$, $A(x)$, $A_d(x)$, $g(p)$ and $r_d(x)$.
any hints will be ...
- 105k
11
votes
2
answers
1k
views
Does the Prime Number Theorem have anything to do with Erdos-Kac law or vice versa?
The prime number theorem says on average we can find $\frac n{\log n}$ primes of magnitude $n$.
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ primes.
Somehow the fact $e^{\...
- 4,713
12
votes
2
answers
616
views
Are there any notion of 'almost primes' known to have small gaps?
A notorious question with prime numbers is estimating the gaps between consecutive primes. That is, if $(p_n)_{n \geq 1}$ is the canonical enumeration of the primes, then set $g_n = p_{n+1} - p_n$. It ...
- 105k
2
votes
1
answer
429
views
Writing integers as determinants of matrices with prime entries.
Below are a couple of idle questions that came up one day when I became curious about "matrix factorizations over $\mathbb Z$". Let's start with size $2$: consider the equation $n= ab-cd$ (*), where $...
22
votes
3
answers
2k
views
Understanding Vaughan's Identity
Vaughan's identity https://proofwiki.org/wiki/Vaughan%27s_Identity is a very useful identity in analytic number theory. The identity expresses the von-Mangoldt function $\Lambda(n)$ as a sum of ...
- 105k
46
votes
4
answers
8k
views
Why could Mertens not prove the prime number theorem?
We know that
$$
\sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x)
$$
where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...
- 105k
4
votes
0
answers
262
views
Error term for Vinogradov's three prime theorem
It can be shown that $$\sum_{a + b + c = N}\Lambda(a)\Lambda(b)\Lambda(c) = \frac{1}{2}\mathfrak{S}(N)N^2 + O(N^2\log^{-A} N)$$ for some $1\ll \mathfrak S(N)\ll 1$using the circle method. Are there ...
- 1,890
1
vote
0
answers
206
views
Can the approach followed in this article be used to improve the upper bounds for $H_{k},k>1$?
In http://arxiv.org/pdf/1405.0682.pdf, the author gives a conditional proof of the twin prime conjecture under both a generalized version of the Elliott-Halberstam conjecture and a hypothesis on the ...
- 105k
9
votes
0
answers
414
views
Number of prime factors in a very short interval
Let $k \geq 3$ be a (large enough) integer, let $x \in \mathbb{R}$,
and set $I_x := [x, x + \log^k x]$.
Some believe that for $x$ large enough there exists a prime $n \in I_x$.
Equivalently, there ...
CommunityBot
- 1
3
votes
1
answer
215
views
Density of triple primes
The conjectural density of twin primes is $\frac {c\cdot n}{(\log n)^2}$ at a $c>0$.
Consider integers of form $p,p+1=2^tq,p+2=r$ where $p,q,r$ are primes and $t\geq1$ holds.
Is there any reason ...
- 105k
1
vote
1
answer
153
views
Specializing non-trivial primality tests
Primes $p$ are integers with no factors (composite allowed) in $[1,p]$. There is a polynomial time test for them.
Given an interval $[a,b]$ what is the best way to test given integer $q$ has no ...
- 13.9k
8
votes
1
answer
360
views
How composite $a^n+b$ is?
In connection with this question and its follow-up.
Suppose that $a\ge 2$ and $b\ne 0$ are integers, and $f$ is a monotonically increasing function such that $f(2)>1$, $f(p)\to\infty$, and the ...
- 10.5k
1
vote
2
answers
234
views
Inquiry on the Chebyshev $\theta$ function
Let
$$\theta(x)=\sum_{p\leq x} \log p$$ be the Chebyshev function over primes $p$.
Computational evidence seems to suggest that $\theta(x) < x$ for every sufficiently large $x$.
But is it true ?...
- 30.1k
1
vote
0
answers
67
views
Small solutions to modular hyperbola
Fix $\ell\in\Bbb N_{>1}$ and small $0<\epsilon\ll1$.
Given $r_1,\dots,r_\ell\in(0,1)$ with $\sum_{i=1}^\ell r_i=1$, is it possible to always find $\Omega(p^{\ell\epsilon'})$ solutions $x_i\in\...
- 13.9k
6
votes
1
answer
481
views
Probabilistic Proofs of Key Number-Theoretic Results
Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$.
Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where ...
- 329
12
votes
0
answers
628
views
Sieve bound for prime $k$-tuples
Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by
$$
\mathfrak{S}(d_1, \ldots, ...
- 105k
6
votes
0
answers
233
views
admissible tuples vs. prime tuples
Let $\rho^\ast(x)$ denote the maximal length of an admissible sequence in $[1,x]$, i.e. of a sequence which does not cover all the residue classes modulo any $n\geq 2$. Hensley and Richards (1974) ...
- 105k
7
votes
2
answers
562
views
Upper bound for $p_{n^2} - p_{(n-1)^2}$?
What is the best unconditional upper bound for $p_{n^2}-p_{(n-1)^2}$ such that $p_n$ is the $n$-th prime number?
Asymptotics suggest it's somewhere near $4 n \ln n$, but how to prove this?
Edit: it'...
- 53.3k
32
votes
3
answers
8k
views
Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)
I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949).
One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$,
$$(1) \qquad\qquad \vartheta(x) \...
- 114k
1
vote
2
answers
462
views
A prime number determined by its congruence relation?
Let $p_i$ denote the $i$-th prime number. Is there any "good function" $k(n)$ such that when we know $x_i$ for which $p_n \equiv x_i \pmod{p_i}$ for all $i\leq k(n)$, it is possible to find a unique ...
CommunityBot
- 1
2
votes
2
answers
1k
views
Estimates for Sum of Prime Factors and Number of Prime Factors
Given a positive integer $n$, I've workout out a formula which involves the expression "sum of distinct primes dividing n" minus "number of distinct prime factors of n."
Are there any known ...
- 105k
3
votes
2
answers
315
views
Smallest constant so that there are at least $n/\log_2{n}$ primes between $n$ and a constant multiple of $n$
What is the smallest known $c$ so that for any $n\geq 2$ there are at least $n/\log_2{n}$ primes between $n$ and $cn$ (inclusive)?
The prime number theorem seems to give an asymptotic result so I am ...
- 105k
0
votes
2
answers
193
views
Space of functions f such that the number of primes in $ [x, x+f(x)] $ remains bounded
Given a positive integer $ n $ , let $ S_{b}(n) $ the set of functions $ f $ fulfilling the following conditions :
1) $ f $ is continuous, positive and increasing on $(n,+\infty) $
2) for ...
- 30.1k
7
votes
2
answers
2k
views
Legendre's Constant
In a couple of web pages, I see that Legendre's constant is defined to be $\lim_{n \to \infty} (\pi(n) - (n/\log(n)))$ (for example, here and here).
Actually the first uses $\lim_{n \to \infty} (\log(...
- 17.6k
21
votes
1
answer
1k
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.
...
CommunityBot
- 1
1
vote
1
answer
466
views
Proof of prime gap bound? [closed]
In another question on mathoverflow (What is the best currently proven bounds on prime gaps?) the following bound on the prime gap was quoted:
$G(X)\ll \frac{X^{0.525}}{\log X}$
How do you prove this, ...
- 105k
7
votes
2
answers
438
views
Generalization of Legendre`s conjecture
Legendre`s conjecture states that there is always a prime between $n^2$ and $(n+1)^2$ for every natural $n$.
It is natural to create following generalization:
Is it true that for every $\...
- 105k
2
votes
2
answers
338
views
Weak form of Brocard's conjecture
I ask this out of curiosity, motivated by a question asked by one of my students.
The Brocard's conjecture claims that there exist at least four prime numbers between $p_{i}^2$ and $p_{i+1}^2$, where ...
- 6,984
1
vote
0
answers
172
views
Possible monotone decreasing sequence involving primes
I am working with the following sequence involving primes $$T_{\alpha}(p_n) = p_n^{\alpha} \prod_{i=1}^{n} \left( 1 - \frac{1}{p_i^{\alpha}}\right)$$ with $\alpha \in (0,1)$. It has been shown Prime ...
- 251
3
votes
1
answer
230
views
Double max of a particular sum in Montgomery-Vaughan
In the Montgomery-Vaughan's paper ''The exceptional set in Goldbach's problem'',
they estimate the following sum:
$$\displaystyle \max_{0<y\leq x}\max_{0<h\leq x} \left(h+\frac{x}{P}\right)^{-1}...
8
votes
1
answer
1k
views
Ambiguity in Nicolas' criterion for the Riemann Hypotheis?
A result of Sole, Planat and Omar's paper, ''Quantum mechanics and the Riemann Hypothesis'', (Theorem 2 with $b=2$), says the RH is equivalent to the statement that for every large enough integer $k$, ...
- 114k
0
votes
1
answer
326
views
Prime numbers property. A Merten's third theorem like sequence
Here is a question I have asked on Math Stack Exchange https://math.stackexchange.com/questions/2290917/prime-numbers-property-mertens-theorem-related-sequence , that I would like this community to ...
- 105k
3
votes
1
answer
224
views
PNT analog for primes inside a structured set
Let $\Bbb T$ be the set of all square free integers with ordering derived from $\Bbb N$. Essentially $PNT$ says if you pick $\log N$ integers less than $N$ you can expect one of them to be prime.
...
- 13.9k
0
votes
0
answers
230
views
Factorial : Gamma :: Primorial :?
Is there a unique function with the following properties:
f is meromorphic on the complex plane;
f is log-convex for n ≥ 1
$f(n) = n\#$ for n prime and ≥ 2, where # is the primorial function, and $f(...
- 309
2
votes
1
answer
267
views
Enquiry on an inequality involving the sum of the reciprocals of primes
Let $p_k$ denote the $k-th$ prime. Do there exist some constant $A>0$ such that for every sufficiently large $k$,
$$\sum_{p\leq p_k} \frac{1}{p} > B + \log\log p_k + \frac{A}{\log p_k}$$
...
- 17.6k
7
votes
1
answer
382
views
$\log \log p / \log \log n$, where $p|n$, gets equidistributed in [0,1] (for almost all $n$)
According to Hardy-Ramanujan/Erdős-Kac we know that usually there are $\sim\log\log n$ prime numbers in a factorization. But if you pick up a natural number at random, and you factor it, what is the ...
- 105k
18
votes
3
answers
2k
views
A question on the prime divisors of p-1
For each positive integer n we may define the convergent sum $$ s(n)=\sum_{p}\frac{(n,p-1)}{p^2} $$
where the summation is over primes p and $(a,b)$ denotes the greatest common divisor of a,b.
It is ...
- 3,062
25
votes
2
answers
3k
views
Prime square offsets: Why is +7 more frequent than -7?
For a prime $p$, define $\delta(p)$ to be the smallest offset $d$
from which $p$ differs from a square:
$p = r^2 \pm d$, for $d,r \in \mathbb{N}$.
For example,
\begin{eqnarray}
\delta(151) & = &...
- 148k
3
votes
1
answer
356
views
Squarefree values of polynomials at prime arguments
This is a reference request.
Assume that $f_1,\ldots,f_r \in \mathbb{Z}[t]$ are non-zero linear polynomial.
Letting $\mu$ be the M\"{o}bius function, is there any work on
$$ \sum_{p\leq x} \prod_{i=...
- 26.9k
4
votes
2
answers
682
views
Is there always at least one prime in intervals of this form?
Take some 4 consecutive primes $p_n,p_{n+1},p_{n+2},p_{n+3}$ where $p_n \geq 5$.
Now form two products: $p_n \cdot p_{n+3}$ and $p_{n+1} \cdot p_{n+2}$.
Is there always at least one prime in the ...
- 105k
2
votes
0
answers
107
views
Maximization of product over primes
I have the following maximization problem. Let $f(p)$ be a real function on the primes, having values in $(0,1)$. Assume that $Y$ is a given (large) positive number, and that we have the bound
$$\...
- 3,432
12
votes
3
answers
929
views
Mertens-like sum in arithmetic progressions
I find myself needing a good estimate for $\sum_{p\le x,\, p\equiv a\bmod q} 1/p$, perhaps something like
$$
\sum_{p\le x,\, p\equiv a\bmod q} \frac1p = \frac{\log\log x}{\phi(q)} + b(q,a) + O\big(\...
- 39.9k
0
votes
0
answers
110
views
Bound for $|p_n - \operatorname{li}^{-1}(n)|$
It is well-known that $|\pi(x)-\operatorname{li}(x)| \leq \epsilon(x)$, where $\pi(x) = \sum \limits_{p \leq x} 1$ is the prime counting function, where $\operatorname{li}(x) = \int \limits_{2}^{x}\...
CommunityBot
- 1
4
votes
1
answer
234
views
Shifted primes avoiding a set of divisors
Let $B$ be a set of positive integers such that $\sum_{b \in B} 1 / \varphi(b) < +\infty$, where $\varphi(\cdot)$ is the Euler's totient function. For any $y > 0$ put
$$\delta_y := \limsup_{x \...