All Questions
Tagged with pr.probability nt.number-theory
181 questions
0
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337
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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 ...
- 2,480
0
votes
1
answer
426
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
$$F(s)=\...
- 2,480
24
votes
2
answers
1k
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 $...
- 551
6
votes
0
answers
671
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 ...
- 2,029
13
votes
4
answers
1k
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 ...
6
votes
2
answers
461
views
Intrinsically measurable subsets of amenable semigroups.
This question is related to the one in https://mathoverflow.net/questions/65322/the-structure-of-certain-maximal-sets-of-means-into-amenable-semigroups. I open a different topic because they can be ...
- 6,034
3
votes
1
answer
845
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: http://en.wikipedia.org/wiki/...
- 26.9k
0
votes
3
answers
293
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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 ...
- 6,034
5
votes
1
answer
1k
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 ...
- 4,952
18
votes
3
answers
918
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 ...
- 82.7k
66
votes
4
answers
4k
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$ ...
- 25.1k
0
votes
1
answer
999
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}
\...
- 265
0
votes
0
answers
319
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}
\...
26
votes
2
answers
2k
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]| ...
- 19.2k
3
votes
7
answers
4k
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 ...
- 179
8
votes
3
answers
847
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 ...
- 3,217
29
votes
3
answers
3k
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 ...
- 2,200
1
vote
0
answers
225
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 ...
- 1,906
1
vote
2
answers
1k
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 ...
- 13.5k
7
votes
1
answer
643
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 ...
- 4,466
8
votes
3
answers
3k
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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 ...
- 121
18
votes
1
answer
872
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 ...
- 14k
11
votes
2
answers
758
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)...
- 7,320
14
votes
6
answers
2k
views
Density of numbers having large prime divisors (formalizing heuristic probability argument)
I want to prove that the set of natural numbers n having a prime divisor greater than $\sqrt{n}$ is positive.
I have a heuristic argument that this density should be $\log 2$, which is approximately ...
- 7,320
17
votes
13
answers
6k
views
Probability in number theory
I am hearing that there are some great applications of probability theory (or more general measure theory) to number theory. Could anyone recommend some good book(s) on that (or other types of ...
4
votes
3
answers
579
views
Average distance between numbers of the form $2^{a}3^{b}$
I want to order all numbers of the form $2^a3^b$. I need to find the average distance between a random consecutive pair.
For example, in case of a random consecutive pair $2^{n'}$ and $2^{n'+1}$, the ...
- 431
44
votes
5
answers
7k
views
Heuristically false conjectures
I was very surprised when I first encountered the Mertens conjecture. Define
$$ M(n) = \sum_{k=1}^n \mu(k) $$
The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...
5
votes
3
answers
4k
views
Counting lattice points on an n-simplex
Imagine an n-simplex, the solution set for the expression: $a_1$*$x_1$ + $a_2$*$x_2$ + ... + $a_n$*$x_n$ = S, where:
$a_1$ through $a_n$ are positive bounded integers
$x_1$ through $x_n$ are ...
7
votes
2
answers
1k
views
An Expectation of Cohen-Lenstra Measure
The Cohen-Lenstra measure on the set of abelian p-groups assigns $\mathbb{P}(G) = \prod_{i \geq 1} \left( 1 - \frac{1}{p^i}\right) \cdot |\mathrm{Aut}(G)|^{-1} $. Apparently, this is equivalent to ...
- 22.8k
26
votes
5
answers
10k
views
Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?
It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...
- 14k
9
votes
6
answers
3k
views
Primes are pseudorandom?
I've been reading the wonderful slides by Terry Tao and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...
- 15.1k