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6 votes
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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 ...
user5831's user avatar
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5 votes
0 answers
614 views

is there a link with the probabilistic model for prime numbers?

Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$. Let : $$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
Lagrida Yassine's user avatar
4 votes
0 answers
168 views

Is there a relation or formula between the correlations of the nontrivial zeros of the Riemann zeta function and the correlations between high points

Consider an interval of length $(\log T)^{\theta}$ for some fixed $\theta > −1$, around a point $1/2 + i y$ on the critical line where $y\in[T,2T]$ and $T$ is large. How do the correlations between ...
Aftermath 12345's user avatar
3 votes
0 answers
169 views

Why is the smallest (fractional) absolute central moment of a Gaussian distribution almost at $\sqrt{3}/2$?

Let $X$ be a standard normal random variable. What $\alpha$ minimizes $E|X|^{\alpha}$? Numerically, $\alpha$ turns out to be equal to $\sqrt{3}/2-\varepsilon$ where $\varepsilon$ is of the order $10^...
Philipp Ustinoc's user avatar
2 votes
0 answers
118 views

the projection distribution induced by integral points on the sphere

Let $A=\{\mathbf{v} \in \mathbb{Z}^{n}: \|\mathbf{v}\|^2= m \}$ and a fixed $\mathbf{y}\in \mathbb{R}^n$, the norm here refers to the Euclidean norm. Suppose $\mathbf{x}$ is a uniform distribution on ...
constantine's user avatar
2 votes
0 answers
146 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 $\...
Turbo's user avatar
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1 vote
0 answers
87 views

Equidistribution of Frobenius Classes

Let $G$ be a reductive group over $\mathbb{Q}$. Let $K$ be a maximal compact subgroup of $G(\mathbb{C})$. Let $S$ be a finite set of primes. For each prime $p$ not in $S$, let $Frob_p$ be a conjugacy ...
Kledin Dobi's user avatar
1 vote
0 answers
78 views

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 ...
user2284570's user avatar
1 vote
0 answers
169 views

Normal numbers and law of the iterated logarithm

If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...
Vincent Granville's user avatar
1 vote
0 answers
123 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 $\...
Turbo's user avatar
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1 vote
0 answers
360 views

Incredibly accurate recursions for the Riemann Zeta function

Last update as of Jan 27, 2021: I posted this as an article for laymen, here. It is very light mathematically speaking, but section 3 is a little more accurate than my post here. During some ...
Vincent Granville's user avatar
-3 votes
0 answers
136 views

Approximation on Dirichlet's arithmetic progression by means of central limit theorem

In this video lecture on Number theory over function fields taught by Will Sawin is presented a 'conceptional' reason for error estimation $\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \} =\frac{1}...
JackYo's user avatar
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