Questions tagged [probabilistic-number-theory]
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68 questions
7
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Distribution of $\int_0^1 \frac{d\sigma}{|\zeta(\sigma+i t)|}$ (or $\max_{0\leq \sigma\leq 1} \frac{1}{|\zeta(\sigma+i t)|}$)
It follows from Selberg's famous distribution result (see this post from Terry Tao's blog for an informal exposition) that $1/\zeta(\sigma+it)$ is bounded for a proportion tending to $1$ (as $T\to \...
- 22k
11
votes
3
answers
909
views
Uniform distribution of sequence mod 1
Is it known whether "for most $r$" the sequence $$r \cdot 2^k \bmod 1, \qquad k \in \mathbb N $$ is uniformly disributed in $[0,1]$?
- 235
1
vote
0
answers
118
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In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?
Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements:
$\lambda$ being a random large prime such as $w^\lambda > 2\times m$
$1 < n < m−1$.
m is ...
- 129
3
votes
1
answer
216
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Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency
let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\} $$
I'm studying fractal geometry and ...
4
votes
1
answer
311
views
Asymptotic density of an infinite union of subgroups
Let $1 < a_1 < a_2 < a_3 <{} ...$ be a sequence of integers. For a subset $A \subset \Bbb Z$, denote by $d(A)$ its natural density (if it exists).
Is it true that $$
\lim_{N \to +\infty} ...
- 275
2
votes
0
answers
104
views
Twin prime distribution centering twice a semiprime
What is the conjectured distributional behavior of semiprimes $pq$ ($p$ and $q$ are primes) having the property $2pq+1$ and $2pq-1$ are primes?
- 1
1
vote
0
answers
95
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Distribution of number of prime factors of $p^k\pm1$
What is the behavior of number of prime factors of integers of form $p^k\pm1$ where $p$ is a fixed odd prime or $2$ and $k$ varies over positive integers?
- 1
1
vote
1
answer
155
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Why is $\sum_{m=1}^{n}\frac{(\nu(m)-\log\log n)^2}{n\log\log n}=\int_{-\infty}^{\infty}\omega^2\, \mathrm{d}\sigma_n(\omega)$?
I was reading an article on Probabilistic Number Theory by M.Kac where I am not able to understand why a particular equation mentioned here in page $657$ equation $(7.7)$ is true?
I do understand that ...
- 309
3
votes
0
answers
144
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Choose a sub series of a random series, such that its expectation can be a given real number
Suppose $a>0$, and we have an infinite series of Bernoulli random variables $B_k$ with
$$\mathbb{Pr}{\large[}B_k=1{\large]} = \frac{1}{1+e^{a\cdot 2^k}}$$
Then
$$\text{E}\left[\sum_{k=-\infty}^{\...
- 43
2
votes
1
answer
606
views
Is there a Cramer's conjecture for Sophie Germain primes?
A prime $q$ such that $2q+1$ is also a prime is a Sophie Germain prime.
Cramer's conjecture tells gap between consecutive primes is bound by $O(\log^2p)$.
Is there a similar conjecture for Sophie ...
- 1
1
vote
0
answers
151
views
On probability of coprimality of a list of numbers
We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$.
Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...
- 1
4
votes
1
answer
430
views
Is there an Erdős–Kac theorem for number of divisors?
Erdős–Kac theorem gives average number of prime factors of an integer.
Is there a theorem which concerns average number of divisors of an integer?
- 1
6
votes
2
answers
2k
views
Freeman Dyson's approach to string theory [closed]
Context:
In celebrating the centenary of Ramanujan's birth, Freeman Dyson presented the following career advice for talented young physicists [1]:
My dream is that I will live to see the day when our ...
- 4,051
2
votes
1
answer
176
views
Is $\lim_{s \rightarrow 1} E(f(X_s)) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N f(k)$?
Let $s>1$ be a real number. We look at the zeta probability function / Zipf probability function defined as:
$$P(X = n) = \frac{1}{n^s \zeta(s)}$$
Suppose $f: \mathbb{N} \rightarrow \mathbb{R}$ is ...
2
votes
0
answers
129
views
Primes as expected values?
This is a follow-up question, which is related to the answer of this quesiton: Is there a connection of prime numbers and extreme value theory?
I will duplicate the answer here, so this question is ...
8
votes
2
answers
786
views
Ways of proving normal distribution (with a view towards Selberg's central limit theorem)
Given an random variable $Y:\Omega \to \mathbb{R}$ with finite mean $\mu$ and finite, positive variance $\sigma^2$, let $X = \frac{Y-\mu}{\sigma}$ be the renormalization with mean $0$ and variance $1$....
- 1,744
5
votes
0
answers
260
views
Maximum entropy methods for probabilistic number theory
Might there be a good survey paper on the application of maximum entropy inference for non-trivial problems in probabilistic number theory?
So far I am aware of the work of Ioannis Kontoyiannis, an ...
- 4,051
5
votes
1
answer
413
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Erdős–Kac and Poisson
This Wikipedia article tells us that the Erdős–Kac theorem says that if $\omega$ is the number of distinct prime factors of $n$ then:
loosely speaking, the probability distribution of $$ \frac{\omega(...
- 12.9k
6
votes
1
answer
425
views
Almost evenly distributed spherical random vectors
Consider $n$ i.i.d spherically distributed random vectors $z_1 ,\cdots , z_n \sim \text{Unif}(\mathbb{S}^{d-1})$. What is the best lower bound on $n$ for which whp there exists a constant $c>0$ ...
- 315
0
votes
1
answer
219
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Upper bound for an infinite series of Pochhammer Symbol
Let $a_n = \frac{1}{n!}\prod_{i=0}^{n-1} (r+\alpha i)$, for constants $0<r, \alpha<1$. The series is convergent by the ratio test. I want to find the exact value or maybe an upper bound for the ...
- 31
0
votes
1
answer
302
views
How differently would we model the distribution of primes if prime gap is larger?
Cramer's conjecture based on his random model provides prime gaps are bound by $O(\log^2p_n)$ where the gap is between $(n+1)$th and $n$th prime.
How differently would primes be modeled if gaps of $O(...
- 1
2
votes
0
answers
207
views
The uniform “probability” on $\mathbf{N}$: What occurs beyond logarithmic density?
This is a follow-up to Question #47134. There is obviously no uniform probability distribution on $\mathbf{N}$ (or $\mathbf{Z}$); however, using the notion of amenability, you can show that any ...
- 294
3
votes
0
answers
193
views
Kubilius model in higher sieve dimension?
The Kubilius model, based on the fundamental lemma of sieve theory, let us approximate the probability of events depending on the variables $X_p$, $p\leq y$, where $X_p=1$ if $p|n$ ($n$ a random ...
- 22k
0
votes
0
answers
193
views
On the $\mathsf{LCM}$ of a set of integers defined by moduli of powers
For integers $a,b,t$ define $$\mathcal R_t(a,b)=\{q\in\mathbb Z\cap[1,\min(a^t,b^t)]: a^t\equiv b^t\bmod q\}$$ and $\mathsf{LCM}(\mathcal R_t(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal ...
- 1,846
0
votes
2
answers
263
views
A variant of Turán–Kubilius inequality
Let $\omega(n)$ the number of distinct prime factors of $n$ (counted without multiplicity). A famous consequence of Turán–Kubilius inequality is
$$
\sum_{n\leq x}(\omega(n)-\log\log x)^2=O(x\log \log ...
- 545
9
votes
2
answers
765
views
Is there a connection of prime numbers and extreme value theory?
As most others are, so am I fascinated by primes.
By the theorem of Euclid and the sieve of Eratosthenes the $ k \ge 2$ - th prime is given by:
$$ p_k = \min_{x>1,\gcd(x,p_1 \cdots p_{k-1})=1} x ...
user6671
4
votes
1
answer
251
views
Small linear relations between primitive Pythagorean triples $\mathsf{II}$
WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$.
Now let $a^2+b^2=c^2$ be a primitive Pythagorean ...
- 1,846
1
vote
1
answer
247
views
Small linear relations between primitive Pythagorean triples $\mathsf I$
Say $a^2+b^2=c^2$ is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation
$$ua^2+vb^2+xab+ybc+zca=0$$ where $(u,v,x, y, z)\in\mathbb Z^4$ are variables. If $(u,v,x, y, z)\neq(...
- 1,846
2
votes
0
answers
192
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Funny questions about Moebius Function
I need to firstly claim that my research is not about number theory, however, I am pretty interested in it, especially funny questions in number theory, e.g. Kollatz Conjecture. Three years ago, I ...
- 61
0
votes
1
answer
171
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Probabilistic interpretation of square free numbers and other properties
We can use the Lindberg condition to show the distribution of number of prime divisors of an integer approaches Gaussian.
Is there a similar probabilistic formulation for square free numbers? That is, ...
- 1,846
6
votes
0
answers
228
views
Smooth integers with lower bound on $\omega(n)$
Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
- 1,846
2
votes
1
answer
185
views
Non-asymptotic results in probabilistic number theory
I'm a beginner. When I searched for results in probabilistic number theory most of the results were asymptotic in nature. Are there any results like with probability 1-$\epsilon$ (w.h.p) some property ...
- 25
3
votes
4
answers
393
views
Approximately satisfying simultaneous vector linear diophantine equations?
Pick three $a,b,c$ vectors in $\mathbb Z^n$ uniformly with $\max(\|a\|_\infty,\|b\|_\infty)<T$ and $\|c\|_\infty<T^2$ and an $\epsilon>0$.
Assume $a$ and $b$ are coordinatewise coprime (...
- 1
0
votes
0
answers
105
views
Deterministically finding a subsequence of integers matching modular roots?
Take two integers $n$ and $m$ with $0<\log_2m<n<m$ and let $r_1=f_1(n)\bmod m$ and $r_2=f_2(n)\bmod m$ for functions $f_1,f_2:\mathbb Z\rightarrow\mathbb Z$.
Denote the $\ell$ roots of $(f_i(...
- 1
1
vote
0
answers
104
views
What is the probability of 'yes' to this likely $coNP$ problem?
Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$.
Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...
- 1,846
1
vote
0
answers
152
views
Probability of small solutions to an uniform random linear diophantine equation?
Take the set $$T(c_1,\dots,c_t)=\{(x_1,\dots,x_t)\in\mathbb Z^t\backslash\{(0,\dots,0)\}:\sum_{i=1}^tc_ix_i\equiv0\bmod q\}$$ where $c_1,\dots,c_t\in(-q/2,q/2)\cap\mathbb Z$.
What is probability ...
- 1
5
votes
1
answer
490
views
How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?
Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...
- 1
1
vote
0
answers
62
views
Does a weighted sequence derived from a sequence matching the Siegel lemma BV bound behave as an uniformly random sequence?
Pick integers $r_1',\dots,r_t'$ that achieve the Bombieri Vaaler bound for the Siegel lemma and find an $m$ (it exists by Dirichlet Pigeonhole) such that given prime $T$ gets $m(r_1',\dots,r_t')\equiv(...
- 1
3
votes
0
answers
181
views
Modular root of $-1$
Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
- 1
-1
votes
2
answers
274
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On distribution of size of integer points in a subspace associated to a linear diophantine equation
Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$
n/4<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|$$ and consider the space of solutions to $ACa+ADb+BCc+BDd=0$ spanned by $3\times 4$ matrix
$$N=...
- 1
3
votes
0
answers
151
views
Improving prime number generation probability?
Deterministic generation of primes in polynomial time is unknown.
Is there a way to probablistically in $O(n^c)$ time bound for some $c>0$ generate polynomially $\Omega(n^c)$ many integers in $[0,2^...
- 1
1
vote
0
answers
166
views
$L^\infty$ norm lower bound for Integer points in null spaces of recursively defined integer vectors?
Letting $\otimes$ be matrix kronecker/tensor product with $n\in\Bbb N$ as a parameter define non-negative integer vectors recursively $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$
...
- 1
1
vote
1
answer
268
views
Generalized notion of divisor function?
Divisor function $d(n,m)$ counts the number of $q\in\Bbb N$ with $b<q<m$ such that $n\bmod q\equiv0$.
Given $b>0$ what is the correct asymptotic, probabilistic and average case behavior of ...
- 1
2
votes
2
answers
774
views
Occurrence of simultaneous small remainders?
Fix $(a,b)=1$, $a<b<2a$ and $a,b>n^{1/(2t)}$ and fix a prime $T\approx n^{\tau+\frac1k}$ where $\tau\geq1$ and $k=2(t-1)$. We can show using exponential sums there is an $m_{_T}$ such that $T/...
user94040
9
votes
2
answers
300
views
If normal with respect to prime base then normal for all bases
I tried to find it on internet but couldn't so m asking this here. I want to ask if a number is normal with respect to all prime number base then do we know that it is normal with respect to any base. ...
- 335
4
votes
0
answers
366
views
Asymptotic estimate for a random model of primes
Question
Let
$$
\pi_{rm_c}(x) = \sum_{ \substack{ {n\leq x}\\{(n+a,P(\sqrt{n}))=1}}} 1-1,
$$
where $P(x)$ is the product of all primes less or equal to $x$ and $a$ is a random integer constrained to ...
- 965
3
votes
1
answer
325
views
Exact statistics in the Frobenius coin problem
The Frobenius coin problem guarantees that if $(a,b)=1$, then
$$ax+by$$ does not represent exactly $\frac{(a-1)(b-1)}2$ numbers all below $g(a,b)=ab-a-b$ if $x,y\geq0$ holds.
Assume $m\in[0,ab-a-b]$ ...
- 1
3
votes
0
answers
266
views
Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 11.4k
4
votes
2
answers
525
views
Relative-totient function (2nd attempt)
Let $\Lambda(x,y)$ be the count of totatives of $x$ that are less than or equal to $y$.
I am asking for the following result to be verified, (particularly the final proposal), I have found no ...
- 161
-3
votes
2
answers
714
views
The number of totatives to the nth primorial, in an interval shorter than the nth primorial
(The notation of this question will be improved over the next few days, sorry for the lack of clarity at the moment.)
Can, and if so when can, we determine the amount of natural numbers which are ...
- 161