All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
4
votes
2
answers
1k
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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 ...
- 25.4k
10
votes
0
answers
512
views
Montgomery's conjecture and lower bound on certain Fourier transform.
Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
CommunityBot
- 1
4
votes
2
answers
699
views
On a sum involving prime numbers
I find myself needing the asymtotics of the following summation for my work. Let $a$ be a positive real number and $p_n$ be the $n$-th prime.
$$
\sum_{k=1}^{n} [k^a - (k-1)^a]p_k
$$
At $a=1$, this ...
- 12.8k
16
votes
4
answers
2k
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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 ...
CommunityBot
- 1
12
votes
1
answer
869
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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 & ...
- 12.3k
6
votes
2
answers
461
views
Density of integers $n$ whose totient $\varphi(n)$ is larger than $\alpha n$
Fix $0 < \alpha < 1$ a real. Let $S_\alpha$ the set of integers $n \geq 1$ such that be $\phi(n)>\alpha n$. For $x>0$, let $S_\alpha(x)$ be the number of positive integers $n$ less han $x$ ...
- 61.9k
12
votes
1
answer
1k
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Prime Power Gaps
In 2000, Baker, Harman and Pintz proved that there is always a prime in
the interval $(n-n^{0.525}, n)$. There are also conditional results
implying smaller intervals. Nevertheless, I could not find ...
- 30.1k
14
votes
1
answer
1k
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Small primes in arithmetic sequences
Fix an integer $a>1$. For $n \geq 1$ an integer, let $\pi_{n,1}(an)$ the number of primes
$p \leq an$ such that $p \equiv 1 \pmod{n}$, and $\pi(an)$ the number of all primes $p \leq an$. Let
$$Q_a(...
- 12.8k
5
votes
1
answer
455
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Large gaps between P2s
Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two ...
- 9,114
6
votes
3
answers
2k
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Better error bounds for partial sums of reciprocals of primes?
One of Mertens' theorems gives that
$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$
where $B$ is a constant near $0.26$ in value and $E(k)$ is an error
term whose size is ...
- 11.4k
7
votes
1
answer
785
views
Logarithmic Integral of n^Zeta Zeroes and certain nested sums of the fractional part function
Below is an approach I've been exploring for connecting the prime counting function with the logarithmic integral and expressing the error term between the two. I find it beguiling, but I've largely ...
- 627
4
votes
1
answer
621
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Progress on Bouniakowsky's Conjecture
Has there been any progress on the Bouniakowsky conjecture? In particular, has anyone been able to prove something for a particular polynomial - or for a class of them?
(I can't seem to find anything,...
- 56.9k
13
votes
3
answers
2k
views
Density of a set of integers
EDIT: this question of mine has received little attention, perhaps in part because it was stated in a too general and complicated way. So let me give it a second chance:
Fix an integer $r \geq 0$. ...
CommunityBot
- 1
19
votes
3
answers
1k
views
Finite sums of prime numbers $\geq x$
Let $S_x$ be the set of finite sums of prime numbers $\geq x$. In other words, let $S_x$ be the submonoid of $(\mathbf{Z}_{\geq 0},+)$ generated by the set $\mathcal{P}_{\geq x}$ of prime numbers $\...
- 326
0
votes
2
answers
515
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Dirichlet's theorem on prime density [closed]
Does anyone knows where I could find the proof of Dirichlet's theorem on the analytic density of primes congruent to a certain integer $m$, which turns out to be $\frac{1}{\varphi(m)}$?
Thanks a lot.
- 9,114
8
votes
2
answers
756
views
The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$
Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a_{n+1}=\operatorname{ GPF}(...
- 83
6
votes
1
answer
826
views
Analogue of van der Corput sequence for prime numbers
A van der Corput sequence is a low-discrepancy sequence over the unit interval first published in 1935 by the Dutch mathematician J. G. van der Corput. It is constructed by placing a decimal point and ...
- 459
5
votes
2
answers
652
views
How are these number-theoretical constants actually distributed?
I'm very curious about this and would be really grateful for any help or comments in this direction. If we consider any of the following number-theoretical constants:
1)The various singular series ...
- 6,198
4
votes
1
answer
928
views
Primes and Ackermann's function
If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all?
EDIT:
I ...
- 1,075
13
votes
0
answers
1k
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Small primes attract large primes
I posted a version of this to stackexchange and got 12 up-votes and no answers in somewhat more than a day. Someone in a comment construed it as asking for a lot of novel research including figuring ...
CommunityBot
- 1
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(...
- 13k
4
votes
1
answer
3k
views
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 ...
CommunityBot
- 1
10
votes
3
answers
3k
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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 ...
- 9,114
11
votes
1
answer
1k
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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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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, ...
- 1,333
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 ...
CommunityBot
- 1
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.$$
What is known about Pierpont primes? I'm not a number theorist, and the best I can find is
http://en.wikipedia.org/wiki/...
- 21
3
votes
1
answer
708
views
Asociated sum series of the Euler Product over the Twin Primes?
Please consider the (presumably infinite) Euler product over the twin primes:
$$ 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((...
- 9,114
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 ...
- 9,114
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 ...
- 1,077
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 ...
- 39.9k
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 $d$. ...
- 9,114
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 ...
- 9,114
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 ...
- 9,114
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 ...
- 31
18
votes
2
answers
2k
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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; ...
- 50.6k
15
votes
3
answers
1k
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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 $\mathbb{C}$?
...
- 50.8k
9
votes
6
answers
3k
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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 ...
- 41.1k