All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
6
votes
4
answers
845
views
Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$
I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click).
In equation (27) the authors, apparently, used the following ...
user40023
8
votes
1
answer
570
views
Asymptotic limit of truncated Legendre sieve
Consider the truncated sum
$$
S(x):=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}\mu(d)/d,
$$
where $P(z)$ is the product of all primes less than or equal to $z$, and $\mu(d)$ is the Möbius ...
- 965
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
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
-4
votes
1
answer
465
views
Asymptotic formula for $\prod_{p\leq x} (1-p^{-1})$ [closed]
Does there exists a good asymptotic formula for
$$A(x) := \prod_{p\leq x}(1-\frac 1p).$$
By using a heuristic argument one can guess:
$$A(x) \sim \frac{1}{2\,\mathrm{ln}(x)}.$$
Here is the argument:...
- 4,484
4
votes
0
answers
117
views
Best constant for Maier's theorem?
Maier proved that, for fixed $\lambda>1,$
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1
$$
and in particular
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\...
- 9,114
21
votes
1
answer
1k
views
Primes that are sums of two squares with constraints on the squares
It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
- 213
4
votes
0
answers
624
views
Is there a hidden symmetry in the prime numbers distribution?
Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
Let'...
- 6,974
0
votes
2
answers
747
views
Has this formula about prime gaps already been conjectured and/or proven?
While playing around with prime gaps, I found out that the following formula seems to be a rather good approximation of the ratio $\dfrac{p_{b}-p_{a}}{b-a}$ where $a<b$ are positive integers:
$$H_{...
- 6,974
2
votes
1
answer
377
views
Prime Number Theorem on APs under various conjectures
I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states:
Unconditionally we have
\begin{equation}
\pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x \...
- 338
5
votes
0
answers
200
views
The divisors of $p-1$ and high-degree residues modulo $p$
Here is a somewhat more explicit version of a question that I asked a while ago.
Suppose that $p$ is a prime of the form $p=2n(n+1)+1$, with a positive integer $n$. Can every odd prime divisor of $...
- 23k
4
votes
1
answer
456
views
References to proofs of upper and lower bounds on the number of coprimes in an interval?
On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
- 965
1
vote
0
answers
216
views
Siegel Walfisz Theorem for algebraic number fields
Is there a generalization of the Siegel Walfisz to algebraic number fields? This has been done for the prime number theorem in the prime ideal theorem.
- 1,890
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 ...
- 3,699
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
4
votes
0
answers
306
views
Effective version of the Bombieri-Vinogradov theorem
Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?
- 1,890
7
votes
1
answer
1k
views
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 ...
- 6,974
8
votes
1
answer
689
views
What does the sum of the reciprocals of all the highly composite numbers converge to?
I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$:
$\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + \...
- 341
3
votes
0
answers
153
views
The behavior of series involving special subsets of the prime numbers
It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ \...
- 3,699
3
votes
1
answer
730
views
what would be the consequences on the distribution of primes of $\Lambda=\infty$?
It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...
- 6,974
6
votes
1
answer
367
views
Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
I have a reference request related to the result :
$\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$
where $P(n)$ is the largest prime factor of the positive ...
- 71
7
votes
2
answers
1k
views
Is there a von Koch-type theorem for the generalized Riemann hypothesis?
Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound
$$
\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).
$$
Q1: ...
- 965
6
votes
0
answers
505
views
$x^2+1$ attaining almost prime values
Iwaniec, using the linear sieve, proved that $n^2+1$ can be a product of at most two primes infinitely often and furthermore a lower bound of the correct order of magnitude for the number of such ...
- 3,062
3
votes
1
answer
860
views
Lower bounds on the error term of the prime number theorem
Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t.
$$f(x)\ll |\psi(x) - x|$$
where $\psi$ is the Chebyshev function.
- 1,890
4
votes
1
answer
980
views
Smallest prime in an arithmetic progression
Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely ...
- 1,890
8
votes
1
answer
605
views
lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?
Are there known any lower and upper bounds for
$$
\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k,
$$
where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$?
Or at least is it known ...
- 841
7
votes
2
answers
426
views
Divisor sums over values of binary forms of primes
Let $\tau$ be the divisor function, that is
$$
\tau(n)=\sharp\{d \in \mathbb{N}, d|n\}.
$$
I was wondering if anyone has ever proved an asymptotic estimate
for the sum
$$S(x):=\sum_{p,q\leq x}\tau(p^...
- 3,062
11
votes
1
answer
700
views
Squarefree numbers $n$ such that $432n+1$ is also squarefree
This is a second attempt (see Primes $p$ such that $432 p +1$ is prime)
Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite?
Fact: the number of such ...
- 123
0
votes
1
answer
461
views
Primes $p$ such that $432 p +1$ is prime [closed]
Is the set of prime numbers $p$ such that $432 p + 1$ is also prime infinite?
It doesn't follow from Dirichlet's theorem as far as I can tell.
- 25
6
votes
2
answers
1k
views
The shortest interval for which the prime number theorem holds [closed]
It is well known that the prime number theorem on the form
\begin{align*}
\pi(x+y) - \pi(x) \sim \frac{y}{\log (x+y)}
\end{align*}
breaks down for short enough intervals, e.g. taking $y=(\log x)^\...
- 965
16
votes
1
answer
1k
views
Tight prime bounds
This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...
- 1,903
6
votes
1
answer
1k
views
Arguments for the second Hardy–Littlewood conjecture being false?
Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that
$$\pi(x + y) - \pi(y) \leq \pi(x).$$
We can easily justify this heuristically, since
$$
\textrm{...
- 965
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
2
votes
0
answers
617
views
Arithmetic progression and average of two prime numbers
Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by:
$$
\ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1.
$$
For all terms of $A$ greater than $\ \...
- 359
5
votes
1
answer
224
views
Log weight removal in general (weaker) prime number theorem
Let $a_n$ be a sequence of non-negative numbers.
Assume that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p\log p}{X}\leq 1.$$
Can we prove that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p}{X/\...
- 131
6
votes
1
answer
382
views
Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$)
Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number
of prime numbers $p \leq n$ in the residue class $r$ (mod $m$).
Further let $1 = r_1 < r_2 < \dots < r_{\varphi(m)}...
- 19.6k
2
votes
0
answers
388
views
Relation between Maier's theorem and a conjecture of Montgomery and Soundararajan
Let us consider the number of primes in the interval $[N,N+h]$, with $h\leq N$. According to the answer given by Lucia to a previous question on the distribution of primes, it is natural to consider ...
- 965
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 ...
- 6,974
7
votes
1
answer
759
views
Major arcs in the proof that every odd number is the sum of at most 5 primes
In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...
- 1,890
3
votes
1
answer
439
views
Wiener-Ikehara tauberian theorem and order of pole at s=1
In the introduction to Akshay Venkatesh's thesis "Limiting Forms of the Trace Formula" we have the following statement :
"For, in summing over primes, the limit
$\lim_{X\to\infty}\frac{1}{X}\...
- 3,799
8
votes
1
answer
508
views
Prime races à la Mertens
I have just read the nice survey by Granville and Martin about prime races.
I wonder what happens if one changes the rules for the prime races as follows.
Fix $q$ a modulus (an integer $>1$). For $...
- 26k
12
votes
1
answer
535
views
Roots of unity near 1 in Z / p Z
Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p.
The ...
- 37k
12
votes
1
answer
1k
views
Does the Maynard-Tao Theorem apply to general tuples of linear forms?
In the paper http://arxiv.org/pdf/1311.5319v1.pdf the author states the following theorem, which he attributes to Maynard and Tao.
For any integer $m > 2$, there exists an integer
$k = k(m)$ such ...
- 303
6
votes
1
answer
727
views
When does Merten's product theorem accurately estimate the number of coprimes in an interval?
Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is:
Let $W(z)=\prod_{p\leq z}\left(1-\frac{1}{p}\right)$. For ...
- 965
5
votes
1
answer
1k
views
Lines in image; are they significant to prime numbers if so how?
Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where $...
- 151
15
votes
1
answer
1k
views
Chebotarev density theorem for $k$-almost primes
Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...
- 26k
4
votes
0
answers
370
views
About sign changes of Li(x)-π(x)
Given a constant $C$, which are the best known upper bounds for the number of sign changes
of the function
$$
f: \mathbb{N} \rightarrow \mathbb{R}, \ \ x \mapsto {\rm Li}(x)-\pi(x)
$$
in the range $...
- 51
10
votes
1
answer
1k
views
Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?
For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq \frac{(2+o(1))x}{\...
- 11.4k
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
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