All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
3
votes
1
answer
516
views
About the asymptotics of LCM
Let $g(x,c)$ be a uniformly random integer in the range $(x,x+c)$ and $LCM[x_1,x_2...x_i]$ the lowest common multiple of the integers $x_i$.
A) Does the limit of (the asymptotics of $LCM[g(3^1,c),g(...
- 61
11
votes
1
answer
1k
views
What might the (normalized) pair correlation function of prime numbers look like?
Cross-posting from Math.Stackexchange.
You might have read about the fortuitous meeting between Montgomery and Dyson. The background is that the nontrivial zeros of the Riemann zeta function, when ...
- 441
8
votes
3
answers
1k
views
Asymptotics for primality of sum of three consecutive primes
We consider the sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where $\{p_i\}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc..
The first few values of $R_n$ are:
10, 15, 23, 31, 41, 49, 59, ...
- 93
21
votes
3
answers
6k
views
Why is the Chebyshev function relevant to the Prime Number Theorem
Why is the Chebyshev function
$\theta(x) = \sum_{p\le x}\log p$
useful in the proof of the prime number theorem. Does anyone have a conceptual argument to motivate why looking at $\sum_{p\le x} \log ...
- 1,533
2
votes
0
answers
292
views
Prime divisors of the difference set
Fix $c\in(0,1)$, and let $N$ be a (large) positive integer. Given a set $A=\{0=a_1<\dots<a_n=N\}$ of density $\alpha:=n/N>c$ with $\gcd(A)=1$, I want to find a prime dividing as few ...
- 23k
4
votes
1
answer
708
views
Calculating the constant in the Bateman-Horn-Stemmler conjecture
Bateman & Horn [1], building on Bateman & Stemmler [2], give a conjectured formula for the density of numbers that produce simultaneous primes in a number of fixed polynomials.
The constant ...
- 9,114
4
votes
2
answers
323
views
Multiplicity one prime in the factorisation of p-N
I'm wondering if analytic number theorists can prove results which have the following flavor:
So let $N$ be a large positive integer.
Q: Can you always find a prime number $p$ in the interval $(N, ...
- 7,609
20
votes
2
answers
1k
views
Median largest-prime-factor
Let $P(n)$ denote the largest prime factor of $n$. For any integer $x\ge2$, define the median
$$
M(x) = \text{the median of the set }\{P(2), P(3), \dots, P(x) \}.
$$
Classical results of Dickman and ...
- 12.8k
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 ...
- 26.9k
17
votes
2
answers
1k
views
Chen's Theorem with congruence conditions.
I would like to revisit a closed question of asterios in a more MO kind of way,
because it cuts quite close to a related question about sieving that might be of general interest.
The original ...
user631
12
votes
1
answer
867
views
Analytic lower bounds on the first sign change of pi(x) - li(x)?
There have been many results on the first sign change of $\pi(x)-{\mathrm{li}}(x)$: among others, Lehman, te Riele, Bays & Hudson, Demichael, Chao & Plymen, and most recently Saouter & ...
- 9,114
2
votes
0
answers
318
views
Pierpont primes
A Pierpont prime is a prime $p$ that can be written as $$p=2^u 3^v + 1.$$
What is known about Pierpont primes? I'm not a number theorist, and the best I can find is
http://en.wikipedia.org/wiki/...
- 21
5
votes
3
answers
3k
views
Asymptotics of Product of consecutive primes
I am looking for the asymptotic growth of product of consecutive primes. Is there anything that is known about this growth?
- 13.9k
3
votes
1
answer
708
views
Asociated sum series of the Euler Product over the Twin Primes?
Please consider the (presumably infinite) Euler product over the twin primes:
$$ f(z) = \prod_{p\in\mathbb{P}}^{\infty} \Big( 1 - \frac{1}{(p(p+2))^ z} \Big) $$ (in which $p(p+2)$ is a divisor of $4((...
- 4,829
7
votes
0
answers
709
views
"probabilistic" density of primes?
A certain set $\cal P$ of primes is defined by two assumedly independent conditions:
The first condition on a prime $p$ can be characterized in terms of the type of splitting of $p$ in certain Galois ...
- 797
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
0
votes
1
answer
284
views
Do the roots of R(x) have any significance for the prime counting function?
I'm calculating the roots of the function
\begin{equation}
R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k})
\end{equation}
This function seems to have a largest and smallest positive root. Can ...
- 3,217
5
votes
2
answers
751
views
Proof in the literature of an equality involving the prime counting function
Let
\begin{equation}
R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k})
\end{equation}
where $\mu$ is the Mobius function and
\begin{equation}
li(x) = \int_0^x \frac{dt}{\log t}
\end{equation}
Is ...
- 3,217
4
votes
2
answers
577
views
What does the probabilistic model suggest the error term in the PNT should be?
Let $\Lambda(n)$ be the von Mangoldt function. The prime number theorem is equivalent to the statement that $\sum_{n \leq N} \Lambda(n) \approx N$. Defining $\lambda_{*}(n)= \Lambda(n)-1$ we may ...
- 13k
4
votes
2
answers
1k
views
Product over the primes
I'm trying to estimate the product
$$\prod_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$
where $p,q,r,s$ are primes.
This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea ...
- 9,114
25
votes
7
answers
3k
views
Question on consecutive integers with similar prime factorizations
Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le e_k$...
- 15.4k
4
votes
2
answers
1k
views
Calculating the infinite product from the Hardy-Littlewood Conjecture F
The Hardy-Littlewood Conjecture F [1] involves the infinite product
$$\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$$
where $\varpi$ ranges over the odd primes and $\left(\frac D\...
- 9,114
42
votes
2
answers
9k
views
Is Li(x) the best possible approximation to the prime-counting function?
The Prime Number Theorem says that $\lim_{n \to \infty} \frac{\pi(n)}{\mathrm{Li}(n)} = 1$, where $\mathrm{Li}(x)$ is the Logarithm integral function $\mathrm{Li}(x) = \int_2^x \frac{1}{\log(x)}dx$. ...
- 5,530
4
votes
1
answer
3k
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Is there another proof for Dirichlet's theorem? [duplicate]
Possible Duplicate:
Is a “non-analytic” proof of Dirichlet’s theorem on primes known or possible?
Dirichlet's theorem on primes in arithmetic progression states that there are ...
- 479
16
votes
4
answers
2k
views
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 $\...
- 9,114
62
votes
1
answer
14k
views
Is the Green-Tao theorem true for primes within a given arithmetic progression?
Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.
Now, consider an arithmetic progression with starting term $a$ and common difference $d$. ...
- 3,699
10
votes
3
answers
3k
views
A number encoding all primes
This may be a soft question, but it's just something I thought of one night before sleeping. It's not my field at all, so I am just asking out of curiosity. Has anyone studied the number which is the ...
- 15.5k
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(\...
- 12.8k
7
votes
4
answers
1k
views
Reference for the expected number of prime factors of n larger than n^alpha is -log alpha
Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$.
It's my understanding that (...
- 14k
69
votes
4
answers
14k
views
Is a "non-analytic" proof of Dirichlet's theorem on primes known or possible?
It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} \}...
- 118k
30
votes
3
answers
4k
views
Heuristic argument for the prime number theorem?
Here is a bad heuristic argument for the prime number theorem. Let $n$ be a positive integer and assume that PNT holds up to $n$. Then $n$ itself is prime if and only if for each prime $p<n$ the ...
- 29k
20
votes
4
answers
3k
views
Primes $p$ for which $p-1$ has a large prime factor
What are the best known density results and conjectures for primes $p$ where $p-1$ has a large prime factor $q$, where by "large" I mean something greater than $\sqrt{p}$.
The most extreme case is ...
- 7,320
18
votes
2
answers
2k
views
Primes of the form a^2+1
The fact that the Riemann zeta function $\zeta(s)$ and its brethren have a pole at $s=1$ is responsible for the infinitude of large classes of primes (all primes, primes in arithmetic progression; ...
- 32.6k
7
votes
2
answers
564
views
Smallest k-term AP of primes
Let $S(k)$ denote the smallest integer such that there exists a k-term arithmetic progression of primes among the integers $[1,S(k)]$. Green and Tao have an unpublished note that gives a very large ...
- 13k
8
votes
1
answer
721
views
Integral of the error estimate in the prime number theorem
This seems like something that should be in discussed in the literature, but I can't find anything. Here $\pi(x)$ is the prime counting function and $\psi(x)$ is the usual sum of the Von Mangoldt ...
- 13k
8
votes
4
answers
1k
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Prime numbers $p$ not of the form $ab + bc + ac$ $(0 < a < b < c )$ (and related questions)
If we ask which natural numbers n are not expressible as $n = ab + bc + ca$ ($0 < a < b < c$) then this is a well known open problem. Numbers not expressible in such form are called Euler'...
- 3,463
15
votes
3
answers
1k
views
Does there exist a meromorphic function all of whose Taylor coefficients are prime?
More precisely, does there exist an unbounded sequence $a_0, a_1, ... \in \mathbb{N}$ of primes such that the function
$\displaystyle O(z) = \sum_{n \ge 0} a_n z^n$
is meromorphic on $\mathbb{C}$?
...
- 118k
9
votes
6
answers
3k
views
Primes are pseudorandom?
I've been reading the wonderful slides by Terry Tao and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...
- 15.1k
9
votes
1
answer
2k
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The large sieve for primes
Let $\Lambda(n)$ be the von Mangoldt function, i.e., $\Lambda(n) = \log p$ for $n$ a prime power $p^k$ and $\Lambda(n) = 0$ for all $n$ that not prime powers. Let
$$S(\alpha) = \sum_{n \leq N} \...
- 20.2k
16
votes
4
answers
2k
views
Arithmetic progressions without small primes
The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :
Is it known that there are infinitely many primes p for which ...
- 156k