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-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}...
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 ...
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 ...
3 votes
1 answer
401 views

Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$

Using my computer, I found that in the interval $[1, N]$ the probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$ is greater than constant $c$ where $N=10^2, 10^3,...,10^{9}$, $x$ ...
6 votes
1 answer
809 views

Probability that a positive integer is in the range of the Euler phi function

Define $f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|$. Clearly, $f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1$ since $\phi(n)$ is even for all $n > 2$. Is $\limsup_{n\...
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 ...
3 votes
2 answers
250 views

Proof of an asymptotic formula by Tricomi

Firs of all I ask my question, then I explain how this question arises in my mind and lastly what I tried to solve it. QUESTION: Let $P_{n,N}(k)$ be the number of composition of an integer $k$ in $n$ ...
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 ...
2 votes
1 answer
159 views

Is the density of integers $n$ such that the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ is surjective positive?

Let $\omega(m)$ be the number of prime factors of $m$ regardless of multiplicity. I'm interested in the behavior of the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ for a given integer ...
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 $\...
8 votes
2 answers
671 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 $1$....
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 ...
5 votes
2 answers
873 views

Exact formula for partial sums of Liouville function $L(n)$ (OEIS sequence A002819)

I am wondering if it is possible to get a useful exact formula, or at least some useful asymptotics, for the partial sums of the Liouville function (OEIS sequence A002819) $$L(n)=\sum_{k=1}^n \lambda(...
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 ...
12 votes
2 answers
820 views

Size of largest square divisor of a random integer

Let $x$ be an integer picked uniformly at random from $1 \ldots N$. Write $x = r^2 t$ where $t$ is square-free. How does the expected value of $r$ scale with $N$? Is anything known about the variance ...
11 votes
1 answer
1k views

Smooth functions that resemble random walks

If the Riemann hypothesis holds, then the Mertens function $M(n)\equiv\sum_{x\leq n} \mu(n)$ behaves much like a 1D random walk. This includes the statements that $M(n)$ changes sign infinitely often ...
1 vote
2 answers
597 views

Moments of complex random variables

My question is that how much information we can get form integer moments of a complex random variable? Let $\mathcal{Z}$ be a complex value random variable, and assume that we can compute $$\int \...
0 votes
1 answer
138 views

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,...
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 \...
3 votes
1 answer
134 views

frequence of block of digits in Mobius sequence

Let $\mu$ be the Mobius function from $\mathbb{N}$ to $\{-1, 0, 1\}$. It is well known for the frequency of $-1, 1$, and $0$ for the sequence $(\mu(1), \mu(2), \mu(2), \dots, )$. For any $k\in \...
7 votes
1 answer
465 views

A theorem by Harald Cramér?

In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement: Suppose $\{X_n\}_{n=2}^\infty$ is a sequence of ...
3 votes
1 answer
315 views

Where to find the proof of this property?

I am doing some exercises in the analytic and there is a problem as following: ``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that: $\sum\limits_{n=1}^{+\infty} f_n = 1$. $\...
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^...
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]| ...
23 votes
2 answers
2k views

Parity of $\lfloor 1/(x y) \rfloor$ not equally distributed

A curious puzzle for which I would appreciate an explanation. For $x$ and $y$ both uniformly and independently distributed in $[0,1]$, the value of $\lfloor 1/(x y) \rfloor$ has a bias toward odd ...
6 votes
1 answer
481 views

Probabilistic Proofs of Key Number-Theoretic Results

Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$. Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where ...
1 vote
2 answers
191 views

On non-singularity of integer matrices with bounded entries

Given $B>0$ and $n\in\Bbb N$ what is the probability that a given $n\times n$ integer matrix with all entries bound by absolute value $<B$ is non-singular? I am looking for precise scaling.
2 votes
2 answers
370 views

Link between Irreducible Factors and Prime Factors (or Cycles of a Permutation)

In "Anatomy of Integers and Permutations", http://www.dms.umontreal.ca/~andrew/PDF/Anatomy.pdf, Granville gives a calibration of cycles of a permutation and prime factors of an integer. "We know ...
14 votes
3 answers
1k views

On the number of consecutive divisors of an integer

Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq ...
54 votes
4 answers
3k views

When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true. For example, it gives some evidence that there are finitely many ...
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 $\...
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 ...
4 votes
2 answers
577 views

What does the probabilistic model suggest the error term in the PNT should be?

Let $\Lambda(n)$ be the von Mangoldt function. The prime number theorem is equivalent to the statement that $\sum_{n \leq N} \Lambda(n) \approx N$. Defining $\lambda_{*}(n)= \Lambda(n)-1$ we may ...
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 ...