All Questions
Tagged with analytic-number-theory prime-numbers
117 questions
5
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3
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Goldbach conjecture and other problems in additive combinatorics
The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance:
$S = T$ is the set ...
- 3,098
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
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
46
votes
4
answers
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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
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...
- 5,470
11
votes
2
answers
3k
views
least prime in a arithmetic progression
Hello
Here I want to consider the simplest arithmetic progression $n\equiv 1\pmod{q}$ where $q$ is a prime. Is it true that we can find a prime $p\leq q^2$ in this arithmetic progression?
This ...
- 2,508
2
votes
2
answers
489
views
On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I_s$ be defined as above.
Is $I_s$ known to be convergent for any real number $s<1$ ?
user140392
22
votes
4
answers
1k
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Small quotients of smooth numbers
Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
- 2,207
17
votes
3
answers
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A variant of the Goldbach Conjecture
I am asking if this variant of the weak Goldbach Conjecture is already known.
Let $N$ be an odd number. Does there exist prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can ...
- 901
16
votes
1
answer
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Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 3,004
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 ...
- 156k
7
votes
1
answer
786
views
Are there effective small intervals in which primes are dense?
As mentioned in Terry Tao's comment to this question, it is constructively known
that there are primes between sufficiently large cubes. $\:$ According to wikipedia,
"there exists a constant $\: \...
user5810
79
votes
6
answers
11k
views
Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?
Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin ...
- 79.9k
56
votes
1
answer
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A mysterious connection between primes and $\pi$
The Prime Number Theorem relates primes to the important constant $e$.
Here I report my following surprising discovery which relates primes to $\pi$.
Conjecture (December 15, 2019). Let $s(n)$ be ...
- 15.6k
34
votes
7
answers
8k
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Explicit formula for Riemann zeros counting function
I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...
- 371
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2
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What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$
Suppose $m$ is a positive integer. A quantity of interest is
$$
H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right)
$$
The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples ...
- 1,354
17
votes
3
answers
2k
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About the prime divisors of values of polynomials
Let $P$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $\mathscr P_P$ be the set of prime numbers dividing some value $P(n)$ with $n \in \mathbb Z$.
Is it true that $\...
- 3,866
16
votes
1
answer
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Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $
A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale.
I would like ask about the much weaker statement forgetting ...
- 22.8k
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
$$
\sum_{p\le x,\, p\equiv a\bmod q} \frac1p = \frac{\log\log x}{\phi(q)} + b(q,a) + O\big(\...
- 12.8k
12
votes
1
answer
869
views
Analytic lower bounds on the first sign change of pi(x) - li(x)?
There have been many results on the first sign change of $\pi(x)-{\mathrm{li}}(x)$: among others, Lehman, te Riele, Bays & Hudson, Demichael, Chao & Plymen, and most recently Saouter & ...
- 9,114
12
votes
1
answer
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Why do the Maynard-Tao weights work so well?
I am looking for an intuitive reason for why the Maynard-Tao weights work well to capture many primes of the form $n+h_1, \ldots , n+h_k$, where $(h_1, \ldots , h_k)$ is any admissible $k$-tuple.
For ...
- 2,283
12
votes
1
answer
1k
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Prime Power Gaps
In 2000, Baker, Harman and Pintz proved that there is always a prime in
the interval $(n-n^{0.525}, n)$. There are also conditional results
implying smaller intervals. Nevertheless, I could not find ...
- 385
11
votes
2
answers
1k
views
Update for 2015: least prime of form nq+1, with q prime?
I have received a complaint about my 2011 answer
least prime in a arithmetic progression
which, indeed, gives conflicting reports about this:
given a prime $q,$ what can we say about an upper ...
- 25.7k
11
votes
1
answer
700
views
Squarefree numbers $n$ such that $432n+1$ is also squarefree
This is a second attempt (see Primes $p$ such that $432 p +1$ is prime)
Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite?
Fact: the number of such ...
- 123
11
votes
1
answer
1k
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The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$
The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\...
- 11.4k
10
votes
1
answer
469
views
Asymptotic behavior of a certain sum of ratios of consecutives primes
I am looking for the asymptotic growth of the following sum
$$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$
where $p_k$ stands for the prime of index $k$.
Manual computations show, for small values ...
- 1,008
10
votes
1
answer
694
views
Set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers
The question is: does the set of prime numbers $q$ such that $\sum\limits_{p\leq\sqrt{q}}p=\pi(q)$, where $p$ are prime numbers, contain infinitely many elements? You can find the first elements here (...
- 189
7
votes
2
answers
679
views
What is wrong with this deterministic algorithm efficiently generating large primes?
According to PolyMath
(Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in k. ...
- 25.4k
7
votes
1
answer
1k
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Heuristic for Montgomery's conjecture
This is my third question on this site regarding Montgomery's conjecture -- and I apologize
if this is too much -- but I am still not understanding well why this conjecture is believed to be true.
...
- 26k
7
votes
1
answer
1k
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What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?
The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
- 6,984
5
votes
1
answer
434
views
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" ...
user95470
5
votes
1
answer
960
views
There at least 4 divisors of $n-1$ which do not divide $\phi(n)$ if $n$ is a composite of the form $6k+1$
If $n$ is composite then $\phi(n) < n-1$ (Euler's totient function) hence there must be one or more divisors of $n-1$ which do not divide $\phi(n)$. For lack of a better terminology, let us call ...
- 5,470
2
votes
0
answers
422
views
Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,098
2
votes
1
answer
928
views
Is there a formula that can predict the primes in the sequence of ratios of consecutive superior highly composite numbers? : $2, 3, 2, 5, 2, 3, 7,...$
This is the sequence of prime numbers which are the elementary building blocks for the superior highly composite numbers:
$2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 2, 23, ...$
The $n^{th}$ ...
- 341
1
vote
1
answer
327
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Symmetry in Hardy-Littlewood k-tuple conjecture
Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h_1, h_2,\cdots, h_i,\cdots, h_{k-1}=d)$ and $(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},\cdots,h'_i=h_{k-1}-...
- 6,984
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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–...
- 13.3k
36
votes
2
answers
7k
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Why do primes dislike dividing the sum of all the preceding primes?
I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
- 5,470
32
votes
3
answers
8k
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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) \...
- 587
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votes
7
answers
3k
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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
23
votes
1
answer
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Does the average primeness of natural numbers tend to zero?
This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value ...
- 5,470
22
votes
1
answer
852
views
How big can a set of integers be if all pairs have small gcd?
Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be?
If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ (...
- 671
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1
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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 ...
- 3,625
21
votes
1
answer
1k
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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.
...
- 24.7k
21
votes
1
answer
1k
views
Primes that are sums of two squares with constraints on the squares
It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
- 213
20
votes
1
answer
1k
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Possible contemporary improvement to bounded gaps between primes?
In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes
Which brings me to my final remark: where to next in the bounded gaps ...
- 35.5k
19
votes
1
answer
2k
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How many primes can there be in a short interval?
Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$.
What is
$$
\limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \...
- 19.6k
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 $\mathbb{C}$?
...
- 118k
13
votes
4
answers
2k
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Proving Mertens' theorem using the prime number theorem
Mertens' Theorem states that
$$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$
This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number ...
- 21.3k
13
votes
3
answers
1k
views
At what point would an elementary generalization of Bertrand's Postulate be interesting?
I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...
- 1,049
13
votes
2
answers
1k
views
Existence of relative Dirichlet density of primes starting with 1
This question is a duplicate of an existing MO question, but that other MO question has an accepted answer that does not actually answer the question, and I'm not sure how to fix that other than by re-...
- 82.7k
11
votes
1
answer
637
views
Primes such that a given number has very small order
The following came up in (a previous version of) this answer.
Question. Let $a > 1$ be a positive integer, and $f \in \mathbf Z[x]$ a polynomial with positive leading term. Does there always exist ...
- 28.7k