# Tagged Questions

For questions on arithmetic functions, a real or complex valued function $f(n)$ defined on the set of natural numbers.

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### Does existence of mean of arithmetic function imply it is bounded except on a set of density zero? [on hold]

Let $f(n)$ be an arithmetic function whose mean is finite. Is $f$ bounded outside a set of density zero?
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### Computational experiments around the divisor functions, and the matrix $A= \begin{pmatrix} n & \sigma(n) \\ 1 & \sigma_0(n) \end{pmatrix}$

Let for integers $n\geq 1$ the sequence of matrices defined by $$A= \begin{pmatrix} n & \sigma(n) \\ 1 & \sigma_0(n) \end{pmatrix},$$ where $\sigma(n)=\sum_{d\mid n}d$ is the sum of divisors ...
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### Can you analyze this identity involving the sum of divisors function and $rad(n)=\prod_{p\mid n}p$?

Let $\sigma(n)$ the sum of divisors function, then by Mobius inversion formula $$\sigma(n)=n-\sum_{\substack{d\mid n,d<n}}\sigma(d)\mu\left(\frac{n}{d}\right),$$ and since this function is ...
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### Two questions about pseudo equidistributed sequences modulo 1

Let $s_n$ a sequence of positive real numbers such that $$\lim_{n\to\infty}\frac{1}{s_n}=0$$ and $$\lim_{n\to\infty}\frac{s_{[nt]}}{s_n}=t,$$ for every real $t\in[0,1]$. See here, page 4. Question ...
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### $Φ_n$ is Euler group, $n> 2$ is an integer, and $m$ the number of solutions of the equation $x^2 = 1$ in the ring $Z_n$.

Prove $$\prod_{i ∈ Φ_n} i=(-1)^{\frac{m}{2}}$$ Then what becomes this identity if $n$ is a prime number? I know that if $x^2=1$, we pair the number with its inverse modulo $n$ in the product. ...
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### How is countably infinite addition defined

In the axiom of additivity of probability theory, the concept of a countably infinite sum, i. e. the sum of countably infinitely many real numbers, is used. Could someone please tell me how that kind ...
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### An identity involving $[\sigma(n)]^2$

For a positive integer $n$, let $\sigma(n)$ denote the sum of the divisors of $n$. For example, $\sigma(1)=1$, $\sigma(2)=3$, $\sigma(4)=7$, etc. I would like to prove the following identity: For ...
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### What about $\lim_{n\to\infty}\frac{\sum_{k=1}^n s_k\mu(k)}{n}$, for the zeros of Dirichlet eta function $s_k=1+\frac{2\pi k}{\log 2}i$ with $k\geq 1$?
Let for integers $k\geq 1$ the corresponding zeros of Dirichlet eta function of the form $$s_k=1+\frac{2\pi k}{\log 2}i,$$ then we can consider the following puzzle, when we multiply previous ...