Questions tagged [arithmetic-functions]
An arithmetic function is one whose domain is the positive integers and whose range is a subset of the complex numbers. There are a number of important number-theoretic examples.
59 questions with no upvoted or accepted answers
0
votes
0
answers
91
views
Arithmetic expansion of harmonic sum
Note: I have modified the initial question as follows:
Let $w_1, w_2, \ldots, w_d$ be positive weights, and $x_1, x_2, \ldots, x_d$ be positive variables. Now, let us consider the following harmonic ...
0
votes
0
answers
155
views
On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$
The Dedekind psi function is defined for a positive integer $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$
with the definition $\psi(1)=1$. See ...
0
votes
0
answers
180
views
When is $\phi(a^n+b^n+c^n)=0\mod n$?
A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...
user147204
0
votes
0
answers
90
views
Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ with $p_1,p_2\equiv 3\bmod 4$?
let $p_1$ and $p_2$ be positive primes such that $p_1,p_2 \equiv 3\bmod 4$
and $\phi$ is the Euler totiont function , I want to find the Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^...
0
votes
0
answers
143
views
Given $\,m=\prod_k {p_k}^{\alpha_k}\,$ and the function $\,g(m)=\sum_k \alpha_k(p_k-1)^2$, find all solutions of the equation $\,g(2n)=n$
Let's consider the unique decomposition of a natural number $\,m\,$ into its prime factors:
$$\prod_k {p_k}^{\alpha_k}$$
Then, let's define the following arithmetic function (completely additive) $\,g:...
- 1,008
0
votes
0
answers
219
views
On an inequality involving the Lambert $W$ function and the sum of divisors function
Let $W(n)$ be the principal/main branch of the Lambert $W$ function (this is the Wikipedia related to this special function). I was inspired in Robin equivalence to the Riemann hypothesis (see [1]) ...
0
votes
0
answers
94
views
Name and theory of multi-valued functions $F:\mathbb{N}^k \rightarrow \mathbb{N}^l$
In computability theory there are considered mostly single-valued functions $f:\mathbb{N}^k \rightarrow \mathbb{N}$. (Let $\mathbb{N}$ be a placeholder for $\mathbb{N}$ or any initial segment $[0,n]$ ...
- 9,720
0
votes
0
answers
202
views
Representation of a number as a sum of co-prime numbers
Conventions
We will call the following product the canonical expansion of $a$:
$$a = \prod\limits_{p_i\in\mathbb{P}}p_i^{\alpha_i}.$$
If $a$ and $b$ are co-prime we write
$$a \perp b.$$
$\beth$-...
- 251
-10
votes
1
answer
555
views
Arithmetic billiards, prime numbers and the Goldbach conjecture
I've edited the following post on Mathematics Stack Exchange, (now closed, at that date I'm suspended) with identifier 4510963, please let me to know if you've some doubt or I can improve the post.
On ...