All Questions
Tagged with analytic-number-theory prime-numbers
690 questions
2
votes
0
answers
158
views
On the connection between $\pi(x)-Li(x)$ and $\theta(x)-x$
Let $\pi(x)$ be the number of primes $p$ not exceeding $x, \theta(x) = \sum_{p\leq x} \log p$ and $Li(x)$ be the logarithmic integral.
Is it true that
$$\pi(x)-Li(x) = \theta(x) - x + O(x^{1/2}\log^{...
- 1,019
5
votes
0
answers
89
views
Is the ratio of a number to the variance of its divisors injective?
The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...
- 44.4k
0
votes
0
answers
83
views
Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?
I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
- 39.9k
0
votes
1
answer
100
views
On the integral $I_s = \int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$-follow up question
This is a follow up on On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$
According to the answer that i got, $I_s$ is not known to converge for any real $s<1$. But suppose $I_s$ ...
- 105k
0
votes
1
answer
256
views
Lower bound for $\prod_{p\equiv 3 \pmod 4} p^{v_p(n!)}$
What is the best lower bound known for $$\prod_{p\equiv 3 \pmod 4} p^{v_p(n!)},$$ where the product is taken over all the primes(congruent to $3$ modulo $4$) less than or equal to $n$.
- 12.8k
3
votes
2
answers
386
views
Prime divisors of $\prod_{i=1}^n (i^2+1)$
Is it true that for every positive integer $n$ there is a prime $p>n,$ which divides $\prod_{i=1}^n (i^2+1)$ ?
- 23k
7
votes
0
answers
274
views
Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) $?
Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that
$$
\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0
$$
My experimental data for $n \le 6 \times 10^5 $...
- 5,470
2
votes
1
answer
170
views
What about a formula similar than Mill's formula, but producing positive integers without repeated prime factors?
The Wikipedia's article for Prime number shows a known and curious formula for primes from its section Formula for primes, I say the Mills' theorem (please see also the Wikipedia Mills' constant).
...
- 9,501
0
votes
0
answers
128
views
Number of primes skipped by binomial coefficients?
Take $$B(l,n)=\binom{n+l}{n}$$ and $\mathcal P(t)=\{p\mbox{ prime}:p|t\}$.
What is the cardinality of $\mathcal P(B(l,n))$?
What is minimum cardinality of $L\subseteq\{1,\dots,n\}$ such that $$\...
- 13.9k
4
votes
0
answers
273
views
Kaczorowski's Paper on Distribution of Primes
I am looking for a digital copy of the following paper by Jerzy Kaczorowski: ON THE DISTRIBUTION OF PRIMES (mod4)
https://www.degruyter.com/view/j/anly.1995.15.issue-2/anly.1995.15.2.159/anly.1995.15....
- 4,713
4
votes
1
answer
549
views
Sum over reciprocal of primes times coefficient
I would like to show that
$$
\sum_{p\leq x} \frac{1}{p^{1+2/\log x}}\left(\frac{\log\left(x/p\right)}{\log(x)}\right)^2=\log\log x +\mathcal{O}(1)
$$
What I have tried
Since we know that
$$
\sum_{p\...
- 105k
5
votes
0
answers
194
views
Asymptotic expansion for the average of $\omega(n)^2$
Let $\omega(n)$ be the prime factors counting function. I computed that for any $k\geq 0$, there exist certain constants $c_{-1},c_0,c_1,c_2,...c_k$ such that
$$\sum_{n\leq x}\omega(n)^2=x(\log\log x)...
2
votes
0
answers
112
views
Queries on distribution of prime divisors by magnitude?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$.
What is the probability distribution of ...
- 13.9k
3
votes
0
answers
280
views
Magnitude and distribution of largest prime factor?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors.
What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number?
What is ...
CommunityBot
- 1
8
votes
1
answer
575
views
Unstable Integers
There is a question that has been bothering my mind for quite a while now. I will present it and my current thoughts and progress on it.
Let the prime factorization of an integer $n$ be
$$n = p_1^{...
- 311
14
votes
2
answers
739
views
How many zeta zeros are needed to accurately calculate five digits for π(1000000), where π(x) is the prime counting function?
John Derbyshire in his book PRIME OBSESSION says on page 343:
"I’ll round off with a complete calculation of $\pi(1,000,000)$, the
number of primes up to one million, using Riemann’s formula -- ...
7
votes
1
answer
1k
views
A curious prime counting approximation or just data overfitting?
I am not sure, if this is a research problem. If not I will move this question to ME:
Let $\Omega(n) = \sum_{p|n} v_p(n)$, which we might view as a random variable.
Let $E_n = \frac{1}{n} \sum_{k=1}^n\...
- 105k
0
votes
0
answers
114
views
The best error term for the second moment
Let $r_2(n)$ be the number of representations of a positive integer $n$ as a sum of two prime squares, i.e. $n=p^2+q^2$. Consider $S_1(x)= \sum_{n \le x} r_2(n)$ and $S_2(x) = \sum_{n \le x}r_2^2(n)$. ...
5
votes
3
answers
809
views
Positive proportion of logarithmic gaps between consecutive primes
For $x, \lambda > 0$, define
$$S_\lambda(x) := \#\{p_{n+1} \leq x : p_{n+1} - p_n \geq \lambda \log x\} ,$$
where $p_n$ is the $n$th prime number. It is known [1] that an uniform version of the ...
- 114k
4
votes
2
answers
704
views
Estimate related to the Möbius function
I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less or equal than $x$ that are square free and with exactly $k$ primes on its decomposition. ...
- 3,486
6
votes
0
answers
206
views
Primes $p\in(n,2n)$ with $(\frac{-n}p)=-1$
Bertrand's postulate proved by Chebyshev states that for any $x>1$ there is a prime $p$ in the interval $(x,2x)$. In 2012 I considered some refinements of this by imposing additional requirement ...
- 15.6k
2
votes
0
answers
121
views
How to choose a prime p s.t. n-th cyclotomic polynomial splits into as much as possible irreducible polynomials while p is almost constant size?
The reason I ask this question is that cyclotomic polynomial is critical to the construction of lattice-based cryptography. In most of the existing lattice-based cryptographic schemes, $n$ is usually ...
- 33
2
votes
2
answers
450
views
On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function
Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime.
Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...
- 1,018
2
votes
2
answers
412
views
Robin's inequality and the zeros of the Riemann zeta function
Robin showed that if $a\in(1/2, 1]$ is the supremum of the real parts of the zeros of the Riemann zeta function $\zeta(s)$, then $f(x)=\Omega_{\pm} (x^{-b})$, where $b$ is some number on $(a-1/2, 1/2],...
- 1,018
3
votes
0
answers
171
views
Estimating integral of product of terms $\cos(t\log p)$
I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function"
Proposition.
Let $T$ be large and let $n=p_1^{\...
- 199
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 ...
- 7,193
1
vote
0
answers
188
views
Questions on Riemann's explicit formula
If we consider this version of the prime-counting function
$$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$
(with $\pi$ being the normal prime-counting function), then we can write $\...
- 749
20
votes
1
answer
2k
views
Circle $x^2 + y^2 = n!$ doesn't hit any lattice points for any $n$ except for $0$, $1$, $2$ and $6$ or does it?
I stumbled across the following problem in high school:$$
x^2 + y^2 = n!
$$
I tested it within my laptop capabilities, watched a 3b1b video Pi in prime regularities, where he explains how to find the ...
- 1,459
36
votes
2
answers
7k
views
Why do primes dislike dividing the sum of all the preceding primes?
I was investigating primes with the property that the sum of the first $n$ primes is divisible by $p_n$. It turns out that these primes are extremely extremely rare. For primes less than $10^9$, I ...
CommunityBot
- 1
1
vote
0
answers
126
views
How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?
Consider a system of $n$ divisibility conditions on $n$ prime variables:
$$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$
where $a_{i,j}$ are bounded integers. How many solutions ...
- 545
6
votes
1
answer
258
views
How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?
I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that
$$
\prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...
- 2,821
9
votes
0
answers
526
views
Prime character sums
Let $p$ be a (large) prime number, and let $\chi : (\mathbf{Z}/p\mathbf{Z})^{\times} \rightarrow \mathbf{C}^{\times}$ be a Dirichlet character of conductor $p$. We have good estimates on the character ...
- 1,109
1
vote
0
answers
274
views
On Primes in Arithmetic Progressions
I was wondering if the following approach is being attempted to prove the twin-prime conjecture.
Tao and Green proved in their paper (2006), that there are arbitrarily long arithmetic progressions ...
- 1
4
votes
1
answer
954
views
Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture
The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series,
$$\Lambda(m)=\...
- 1,183
3
votes
2
answers
417
views
Infinitely many primes in particular progressions
I'm faced with the following problem on primes. Does someone have any clue? Is it (a reformulation of) an open problem?
Let $d$ be a positive integer, $d\geq 2$. By Dirichlet's theorem, there is an ...
- 12.8k
2
votes
0
answers
76
views
Is there an estimate available for a sum of the form $\sum_{\mathbf{x} \equiv \mathbf{a} (H) } \mu^2(x_1 x_2)$
I am interested in a sum of the shape
$$
\sum_{ \substack{ 1 \leq x_1, x_2 \leq B\\
\mathbf{x} \equiv \mathbf{a} (H) } } \mu^2(x_1 x_2).
$$
I figured it must have been considered before, but I have ...
- 3,625
7
votes
1
answer
501
views
Smallest Mazur's good prime
Let $p$ and $\ell$ be primes $\geq 5$ such that $\ell$ divides $p-1$. Following Mazur, we say that a prime $q$ is a $\textit{good prime}$ if $\ell$ does not divide $q-1$ and $q$ is not a $\ell$th ...
- 12.8k
1
vote
1
answer
144
views
Factoring with partial information on gaps
If $N=PQ$ is a semi-prime with $P=N^{\frac12 +\delta}$ and $Q=N^{\frac12-\delta}$ then if we know $\delta\in(0,\frac12)$ to a reasonable precision we can factor $N$ quickly. What precision (number of ...
CommunityBot
- 1
4
votes
2
answers
555
views
upper bound of consecutive integers which are not coprime with $n!$
Is there any research on getting upper bound of the maximal possible number of consecutive positive integers which are less than $n!$ and NOT coprime with $n!$?
Easy to see that lower bound $\ge n$, ...
- 13k
6
votes
1
answer
1k
views
Arguments for the second Hardy–Littlewood conjecture being false?
Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that
$$\pi(x + y) - \pi(y) \leq \pi(x).$$
We can easily justify this heuristically, since
$$
\textrm{...
- 1
4
votes
2
answers
2k
views
Chebyshev's bias-conjecture and the Riemann Hypothesis
Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?
- 4,713
1
vote
0
answers
66
views
Probability distribution from standard domain (two primes) - IV
Pick a random pair $(a,b)\in\mathbb Z_n^2\setminus\{0,0\}$. Denote $N_2(a,b,n)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
CommunityBot
- 1
10
votes
0
answers
255
views
How many partition values are expected to be prime?
Let $p(n)$ be the partition function. Let $P(N)$ count how many $1\leq n\leq N$ are such that $p(n)$ is prime.
Are there any heuristics for how $P(N)$ should behave?
A crude guess at how this ...
- 7,013
-3
votes
1
answer
269
views
Negative Dirichlet Pigeonhole Principle [closed]
From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|_\infty<\lceil\sqrt p\rceil$ holds where ...
- 13.9k
7
votes
1
answer
508
views
Equidistribution of $\{p_n^2α\}$
Let $p_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p_n \alpha\}$ is equidistributed. Is it known whether the sequence $\{p_n^2 \alpha \}$ is ...
- 7,013
6
votes
0
answers
255
views
Gaussian square-free moat
Is there a sequence $\{z_n\}_{n=1}^\infty$ of distinct square-free
Gaussian integers with $$\sup_{n \geq 1} |z_{n+1} - z_n| < \infty ?$$
For the analogous problem with Gaussian primes instead, ...
- 9,501
6
votes
1
answer
328
views
Reference Request for a result on divisors of $p-1$
I have seen this result in several places without an English reference:
There exist infinitely many primes $p$ such that $p-1=2q_1q_2$ where $q_1$ and $q_2$ are prime numbers with $q_1,q_2>p^{1/4}$...
- 105k
5
votes
1
answer
943
views
Are all the numbers $\pi(n^2)/n^2\ (n=1,2,3,\ldots)$ pairwise distinct?
For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. A well-known conjecture of Legendre states that $\pi(n^2)<\pi((n+1)^2)$ for any positive integer $n$. Here I ask the ...
- 4,713
1
vote
0
answers
156
views
On segments of the series $\sum_p\frac1{p-1}$
Here I ask a question concerning segments of the divergent series
$$\sum_p\frac1{p-1}=\sum_{k=1}^\infty\frac1{p_k-1},\tag{$*$}$$
where $p$ runs over all the primes, and $p_k$ denotes the $k$-th prime.
...
- 15.6k
5
votes
1
answer
999
views
Generalization of Mertens' theorem
One classical Mertens' theorem tells us that $$\prod_{p \leq n} (1-\frac{1}{p})^{-1} = e^\gamma \log n + \mathcal{O}(1).$$
It is now very natural to ask, whether we have some good estimate to $$\prod_{...
- 113