# Questions tagged [probabilistic-number-theory]

The probabilistic-number-theory tag has no usage guidance.

64
questions

4
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### 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} ...

0
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0
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29
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### '$r$-prime factor number' distribution centering an '$l$-prime factor number'

Let '$k$-prime factor integer' $n$ be an integer of form
$$2^t\prod_{i=1}^{k'} p_i^{e_i}$$
where $1\leq k'\leq k$ and at every $i\in\{1,\dots,k\}$ $p_i$ is a distinct odd prime and $e_i\in\mathbb Z_{&...

2
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0
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58
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### 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
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0
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59
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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
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1
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92
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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 ...

3
votes

0
answers

113
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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}^{\...

2
votes

1
answer

480
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### 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
vote

0
answers

112
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### 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 $\...

4
votes

1
answer

291
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### 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?

6
votes

2
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### 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 ...

2
votes

1
answer

122
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### 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 ...

8
votes

2
answers

498
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### 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 $1$....

4
votes

0
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139
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### 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 ...

2
votes

0
answers

84
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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(...

6
votes

1
answer

303
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### 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 $c>0$ ...

0
votes

1
answer

107
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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 ...

0
votes

1
answer

242
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### 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(...

2
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0
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135
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### 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 ...

3
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0
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158
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### 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 ...

0
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0
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150
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### 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 ...

0
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2
answers

162
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### 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 ...

9
votes

2
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656
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### 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 ...

4
votes

1
answer

184
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### 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
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1
answer

165
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### 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(...

2
votes

0
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162
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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 ...

0
votes

1
answer

121
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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,...

6
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0
answers

190
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### 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 ...

2
votes

1
answer

144
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### 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 ...

3
votes

4
answers

278
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### 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 (...

0
votes

0
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83
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### 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
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0
answers

87
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### 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
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0
answers

131
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### 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 ...

5
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1
answer

405
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### 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
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0
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### 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(...

3
votes

0
answers

159
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### 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 ...

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2
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227
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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=...

3
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0
answers

127
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### 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,...

1
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0
answers

128
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### $L^\infty$ 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
vote

1
answer

222
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### 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 ...

2
votes

2
answers

722
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### 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/...

9
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2
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226
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### 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. ...

4
votes

0
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322
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### 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 ...

3
votes

1
answer

177
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### 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 $m\in[0,ab-a-b]$ ...

3
votes

0
answers

225
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### 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 ...

4
votes

2
answers

419
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### 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 ...

-3
votes

2
answers

573
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### 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 ...

3
votes

0
answers

219
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### Probability that an integer contains no $1\bmod 4$ prime factor

$n$ represents integer variable.
What is the probability that and integer contains at most $r(n)$ prime factors of form $1\bmod 4$ where $r(n)$ is a function of $\omega(n)$ (number of distinct prime ...

13
votes

2
answers

771
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### Number of distinct factors

Denote $\omega(m)$ to be number of distinct factors of $m$ as defined in http://mathworld.wolfram.com/DistinctPrimeFactors.html.
At every $c>0$, given $n\in\Bbb N$ define $$S(n,c)=\big\{m\in\Bbb N:...

5
votes

1
answer

492
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### How many random matrices does it take to generate a matrix algebra?

Let $\mathbb{F}$ be a finite field.
Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra.
Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$
that one needs to take ...

18
votes

1
answer

1k
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### How many integers divide a number that involves just three non-zero digits?

Just to be concrete, consider the digits to be binary. Hasse showed that among all the primes, only a fraction of $17/24 < 1$ divide a number of the form $2^n+1$. As a result, the integers that ...