# Tagged Questions

**0**

votes

**0**answers

77 views

### Upper bound for $r_{0}(n)$ through probabilities

Assume Goldbach's conjecture. Then for every integer $n>1$ there exists a non negative integer $r$ such that both $n-r$ and $n+r$ are primes. For a given $N$, let's denote by $r_{0}(N)$ the ...

**7**

votes

**2**answers

271 views

### Sets whose elements are mutually “weakly” coprime?

Fix $n$ and $k$. I want a set $S\subseteq\{1,\ldots,n\}$ with the property that for every $x\in S$,
$$\mathrm{gcd}\bigg(x,\prod_{y\in S\setminus\{x\}}y\bigg)<\frac{x}{k}.$$
How small should a ...

**2**

votes

**0**answers

130 views

### Odds of projections of a point not on the hyperplane

Let $\mathcal{L}=\{\Bbb x\in\Bbb R^n:x_1+x_2+\dots+x_n=0\}$ be a specific hyperplane.
Let a projection of $c\in\Bbb R^n$ be $p(c)=[p_1,p_2,\dots,p_n]$ where $p_i\neq c_i\implies p_i=0$.
Let ...

**2**

votes

**0**answers

128 views

### Fractional Derivatives [closed]

How far these Theories of "Fractional Derivatives" be rigorized ? I have few books and references on Fractional Differential Equations etc (mainly they stress on Applied Mathematics parts and similar ...

**10**

votes

**3**answers

587 views

### Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let
$f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

**2**

votes

**2**answers

222 views

### what's the best way to characterise the distribution of prime elements in simple perfect squared squares

DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...

**1**

vote

**0**answers

140 views

### Bunimovich stadium bouncing ball

http://terrytao.wordpress.com/2008/07/07/hassells-proof-of-scarring-for-the-bunimovich-stadium/
I cannot relate how the eigenfunctions normalizaed correspond to the probabality distribution of the ...

**4**

votes

**2**answers

591 views

### Uniformly distributed sequence in $\mathbb{R}$

We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and
$$\lim_{N \to \infty} ...

**7**

votes

**2**answers

599 views

### Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?

Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , ...

**0**

votes

**0**answers

177 views

### Period of decimal for $1/n$, odd part of $n+1$, and primes.

Let $n$ be a nature number is relatively prime to 10,such the period of the decimal expansion of $1/n$ is $n-1$ or a divisor of $n-1$, and let $c$ be the "cycle length of $n$" (defined below). If ...

**2**

votes

**0**answers

311 views

### Matula-Goebel ordering of rooted trees intrinsic?

I was somewhat recently introduced to the Matula-Goebel bijection between rooted trees and natural numbers. (nicely illustrated here http://keithbriggs.info/matula.html) Looking through them, I ...

**2**

votes

**0**answers

149 views

### Expected period of quadratic generator

I am interested in the mean period of a quadratic congruential generator. Let $X_{n+1} = \sum_{i=0}^2 a_i X_n^i \bmod m$ where the $a_i \in \mathbb{Z_m}$ are chosen uniformly at random and $m$ is a ...

**6**

votes

**2**answers

863 views

### Probability that randomly chosen integers from a restricted set of natural numbers are coprime

We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is
$$
P(k) = \frac{1}{\zeta(k)}.
$$
I am looking at a special case of ...

**1**

vote

**1**answer

371 views

### Probability of all combinations of k numbers among n being coprime

A simple argument shows that if we choose $n$ positive integers at random, the probability of their greatest common divisor being 1 is $1/ \zeta (n)$ (in the sense that if we choose the numbers among ...

**2**

votes

**0**answers

110 views

### Are numbers $h_{r,s} = \sum_{k} P(r;s;k) \frac{1}{n^{2k}} \bigg(1-\frac{1}{n}\bigg)^{n-2k}$ irrational?

I asked this question on MSE and Mike Spivey gave an insightful answer. I decided to put it here nevertheless in case someone else gets interested. If this violates rules on MO, please let me know, ...

**5**

votes

**2**answers

1k views

### What is a random number? (poll experiment) [closed]

Imagine the following experiment: you wait say at a subway exit, and ask everyone passing "please tell me a number" (positive integer, of course). You do this day after day, until you reach say 1M ...

**29**

votes

**4**answers

2k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**2**

votes

**3**answers

782 views

### The probability that a random number N has at least M factors

That is, how to calculate it given the size of N(that is, logN) and assuming that logN is much greater than M. Its an approximation. There is no exact formula.
I do know that according to the prime ...

**2**

votes

**0**answers

164 views

### Small geometric progression modulo N

An problem related to integer factorization using the General Number Field Sieve is the following:
Let $N$ be a composite. Must there exist a 5-term geometric progression $\lbrace ...

**8**

votes

**2**answers

612 views

### Random pseudoprimes vs. primes

(Edit. What I called "pseudoprimes" are known as "CramÃ©r random primes" in the literature,
of which I was unaware.)
Say that a set $S$ of natural numbers is a set of pseudoprimes if they
are (a) ...

**1**

vote

**1**answer

333 views

### Probability that p and q are both prime provided q-p=2r

Hello,
I would like to know whether there is a way, thanks to the prime number theorem, to give some kind of an equivalent of the probability that two positive integers $p$ and $q$ less than a given ...

**1**

vote

**0**answers

195 views

### Calculating or estimating a combinatorial multivariate sum

Dear all,
I'm currently looking at a problem in which the following combinatorial product emerges:
$c(m_1,\dots,m_\lambda;n_1,\dots,n_\lambda)=\frac{m_1 ...

**2**

votes

**0**answers

188 views

### Generator density in $\mathbb{Z}^*_p$

Hello,
Consider the multiplicative group $(\mathbb{Z}/p)^*$ for a given prime $p$. We know that the number of generators in this group is $\phi(p-1)$ --- the Euleur totient function. The question is, ...

**3**

votes

**1**answer

295 views

### accumulation points within Pisot numbers

Recall that Pisot numbers are algebraic integers greater than $1$, whose other Galois conjugates have modulus $<1$. The set of Pisot numbers is usually denoted $S$. It is known that $S$ is ...

**2**

votes

**0**answers

123 views

### finding rank-3 tensors compatible with a rank-2 tensor projection

I am interested in the following problem: Consider a rank-3 symmetric tensor $\boldsymbol{\sigma}$ with $\sigma_{ijk}$ where $\sigma_{ijk}$ can be 0 or 1, and the symmetry is with respect to any ...

**3**

votes

**2**answers

390 views

### using distribution of primes to generate random bits?

In his popular science book The Music of the Primes, Marcus du Sautoy tries to link the truth of the Riemann Hypothesis to the "randomness" of the primes. To do this, he invokes the idea of a "fair ...

**1**

vote

**0**answers

280 views

### What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?

This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...

**0**

votes

**1**answer

292 views

### Lower bounds for partial sums of multiplicative functions

The motivation for this enquiry is to understand something about the impact of multiplicativity for $f:\mathbb{N}\rightarrow\mathbb{C}$ on the conditional convergence of Dirichlet series
...

**1**

vote

**0**answers

214 views

### A probabilistic approach of (a weaker form of) Goldbach's conjecture

Hello,
this message follows on from About Goldbach's conjecture and aims at modelling the quantity $N_{2}(n)$ by a random variable.
Let $n$ be a fixed natural number and let's define for every ...

**21**

votes

**2**answers

868 views

### Drawing natural numbers without replacement.

Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all ...

**6**

votes

**0**answers

519 views

### Is there a probabilistic interpretation of Dedekind zeta functions?

Reading the interesting paper Honest Bernoulli excursions by Smith and Diaconis motivated the question whether probabilistic interpretations for general Dedekind zeta functions are known.
In the ...

**12**

votes

**4**answers

980 views

### What results would follow from or imply “randomness” of the primes?

This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the ...

**5**

votes

**2**answers

402 views

### Intrinsically measurable subsets of amenable semigroups.

This question is related to the one in http://mathoverflow.net/questions/65322/the-structure-of-certain-maximal-sets-of-means-into-amenable-semigroups. I open a different topic because they can be ...

**3**

votes

**1**answer

627 views

### Special case of Duffin-Schaeffer conjecture

The Duffin-Schaeffer conjecture is an old conjecture in metric number theory which has withstood attempts to solve it for about 70 years. The statement can be found here: ...

**0**

votes

**3**answers

234 views

### How can we pave the multiplicative semigroup $(\mathbb N,\cdot)$?

Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s_1,...s_k\in S$ such that the sets $s_i\cdot W$ are pairwise ...

**3**

votes

**1**answer

618 views

### Self Avoiding Walk Enumerations

Let $c(n)$ be the number of Self avoiding walks (SAW) of length $n$ on an infinite lattice $L$. Are there any known non-geometric interpretations of $c(n)$?. For example, is there a number theoretic ...

**18**

votes

**3**answers

765 views

### Can Gauss sums derandomize any heuristic arguments?

I've always been fascinated by the fact that the classical Gauss sum has absolute value $\sqrt p$, which is exactly what we would expect if we were to interpret the Gauss sum as a random walk. In ...

**51**

votes

**3**answers

2k views

### Perron number distribution

A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...

**0**

votes

**1**answer

746 views

### Generalizations of a product formula for the gamma function

Hello and Happy holidays.
I am interested in generalizations of the following product formula for the gamma function
$\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:
\begin{align}
...

**0**

votes

**0**answers

273 views

### Estimating a multinomial sum

I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
...

**20**

votes

**2**answers

1k views

### Is there any sense in which Dirichlet density is “optimal?”

A philosopher asked me an interesting math question today! We know that there are sets S of integers which don't have a "natural" or "naive" density -- that is, the quantity (1/n)|S intersect [1..n]| ...

**2**

votes

**7**answers

1k views

### How to tell if two random polynomials are identical

Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...

**7**

votes

**3**answers

479 views

### Random linear recurrence relations

Problem
I am interested in the random recurrence relation of the form $x_{n+1}=\alpha x_n \pm \beta x_{n-1}$ where $\alpha$, $\beta$ are known constants and the $\pm$ sign is chosen with equal ...

**27**

votes

**3**answers

2k views

### Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...

**1**

vote

**0**answers

186 views

### What is the limiting distribution of local minima of n mod i, for i up to $\sqrt{n}$, as $n \rightarrow \infty$?

The sequence n mod i
Consider the sequence n mod i for i=1...$\sqrt{n}$. If we draw the sequence as an xy-plot, we get a dense triangle (since n mod i < i). More precisely, the limiting density of ...

**1**

vote

**2**answers

657 views

### Sum of digits iterated

Original version.
I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of ...

**6**

votes

**1**answer

453 views

### distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field

This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix ...

**7**

votes

**3**answers

3k views

### Is there any finitely-long sequence of digits which is not found in the digits of pi? [closed]

I know it's likely that, given a finite sequence of digits, one can find that sequence in the digits of pi, but is there a proof that this is possible for all finite sequences? Or is it just very ...

**14**

votes

**1**answer

492 views

### What's the probability that k + n^2 is squarefree, for fixed k?

While playing around with this question (when is the sum of two squares squarefree?), from some experimental computations (and bolstered by the fact that the density of squarefree positive integers is ...

**8**

votes

**2**answers

576 views

### Notions of “independent” and “uncorrelated” for subsets of the natural numbers

In probability/statistics, there is a notion of two things being "independent", which would basically mean that any information we can get about one thing has no effect on our (probabilistic) ...