All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
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
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
56
votes
1
answer
4k
views
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
53
votes
5
answers
4k
views
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
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
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...
- 5,470
45
votes
3
answers
6k
views
Why such an interest in studying prime gaps?
Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions :
lower bounds (recent works by Maynard, Tao et al. [1])
upper bounds (...
- 5,424
42
votes
2
answers
9k
views
Is Li(x) the best possible approximation to the prime-counting function?
The Prime Number Theorem says that $\lim_{n \to \infty} \frac{\pi(n)}{\mathrm{Li}(n)} = 1$, where $\mathrm{Li}(x)$ is the Logarithm integral function $\mathrm{Li}(x) = \int_2^x \frac{1}{\log(x)}dx$. ...
- 5,530
39
votes
1
answer
2k
views
Prime number races in 2 dimensions
Is the mapping $$f: \ \mathbb{N} \rightarrow \mathbb{Z}[i], \ \ \ n \ \mapsto
\sum_{2 < p \leq n \ {\rm prime}} e^{\frac{p-1}{4} \pi i}$$ surjective?
In 1999, when I was an undergraduate student, ...
- 19.6k
36
votes
2
answers
7k
views
Why do primes dislike dividing the sum of all the preceding primes?
I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
- 5,470
34
votes
7
answers
8k
views
Explicit formula for Riemann zeros counting function
I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...
- 371
32
votes
3
answers
8k
views
Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)
I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949).
One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$,
$$(1) \qquad\qquad \vartheta(x) \...
- 587
30
votes
2
answers
4k
views
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
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
28
votes
3
answers
3k
views
Expressing the Riemann Zeta function in terms of GCD and LCM
Is the following claim true: Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,
$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...
- 5,470
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
25
votes
2
answers
3k
views
Prime square offsets: Why is +7 more frequent than -7?
For a prime $p$, define $\delta(p)$ to be the smallest offset $d$
from which $p$ differs from a square:
$p = r^2 \pm d$, for $d,r \in \mathbb{N}$.
For example,
\begin{eqnarray}
\delta(151) & = &...
- 151k
25
votes
7
answers
3k
views
Question on consecutive integers with similar prime factorizations
Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le e_k$...
- 15.4k
24
votes
1
answer
2k
views
Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
- 429
24
votes
1
answer
693
views
Gaussian primes in small boxes
The best unconditional result bounding prime gaps is due to Baker, Harman and Pintz, and states that for any sufficiently large $n$, the interval $$[n,n+Cn^{0.525}]$$ contains a prime, for some ...
- 11.4k
23
votes
1
answer
3k
views
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
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 ...
- 3,625
22
votes
4
answers
6k
views
How does Yitang Zhang use Cauchy's inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum
I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University.
Today I found a detail in the proof ...
- 2,913
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 ...
- 3,625
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
22
votes
4
answers
1k
views
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
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
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
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.
...
- 24.7k
20
votes
1
answer
2k
views
Circle $x^2 + y^2 = n!$ doesn't hit any lattice points for any $n$ except for $0$, $1$, $2$ and $6$ or does it?
I stumbled across the following problem in high school:$$
x^2 + y^2 = n!
$$
I tested it within my laptop capabilities, watched a 3b1b video Pi in prime regularities, where he explains how to find the ...
- 343
20
votes
4
answers
3k
views
Primes $p$ for which $p-1$ has a large prime factor
What are the best known density results and conjectures for primes $p$ where $p-1$ has a large prime factor $q$, where by "large" I mean something greater than $\sqrt{p}$.
The most extreme case is ...
- 7,320
20
votes
2
answers
1k
views
Median largest-prime-factor
Let $P(n)$ denote the largest prime factor of $n$. For any integer $x\ge2$, define the median
$$
M(x) = \text{the median of the set }\{P(2), P(3), \dots, P(x) \}.
$$
Classical results of Dickman and ...
- 12.8k
20
votes
1
answer
1k
views
Possible contemporary improvement to bounded gaps between primes?
In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes
Which brings me to my final remark: where to next in the bounded gaps ...
- 35.5k
20
votes
2
answers
2k
views
Is every prime the largest prime factor in some prime gap?
Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...
- 5,470
19
votes
1
answer
2k
views
How many primes can there be in a short interval?
Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$.
What is
$$
\limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \...
- 19.6k
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 $\...
- 20.8k
18
votes
1
answer
1k
views
The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?
Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...
- 965
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
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
18
votes
1
answer
4k
views
Tightening Zhang's bound [closed]
Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang.
The original bound was $70,000,...
17
votes
3
answers
3k
views
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
17
votes
3
answers
2k
views
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
17
votes
2
answers
2k
views
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 $$x=P_{n+...
- 1,776
17
votes
2
answers
1k
views
Chen's Theorem with congruence conditions.
I would like to revisit a closed question of asterios in a more MO kind of way,
because it cuts quite close to a related question about sieving that might be of general interest.
The original ...
user631
16
votes
4
answers
2k
views
Who first proved that there are at least n^(1-ε) primes up to n?
It's well-known that Hadamard and de la Vallée-Poussin independently proved the Prime Number Theorem in 1896: that $\pi(n)=n/\log n+o(n/\log n)$. I'm curious as to a weaker result: that for any $\...
- 9,114
16
votes
1
answer
1k
views
Tight prime bounds
This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...
- 1,903
16
votes
2
answers
2k
views
Could this unexpected bias in the distribution of consecutive primes have any impact on the security of encryption algorithms?
In a recent paper a quite unexpected result about a new pattern in prime numbers emerged:
Unexpected biases in the distribution of consecutive primesby Oliver, R. J. L.; Soundararajan, K. (Submitted ...
- 5,935
16
votes
4
answers
2k
views
Arithmetic progressions without small primes
The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? :
Is it known that there are infinitely many primes p for which ...
- 156k
16
votes
1
answer
4k
views
Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 3,004
16
votes
1
answer
1k
views
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