# Questions tagged [arithmetic-functions]

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

411 questions
Filter by
Sorted by
Tagged with
13 views

### if such counter example to Lehmer's totient problem exists then could we have more counter examples?

Lehmer's totient problem asks whether there is any composite number $n$ such that Euler's totient function $φ(n)$ divides $n − 1$. which it is unsolved problem or we may reformulate that question as : ...
34 views

### An identity of Arithmetic Functions

Problem: Show that for all positive integers $n$, $$\sum_{a=1, (a,n)=1}^{n} (a-1, n) = d(n)\phi(n)$$ where $(a, b)$ stands for $\text{gcd}(a, b)$ and $d, \phi$ are the divisor and Euler's totient ...
28 views

### Summation of certain divisible numbers

Let $C(N,m)$ be the number of positive integers $\le N$ which are relatively prime to $m$. It can be found by following equation \begin{align} C(N,m)=\sum_{d|m}\mu (d)\left\lfloor\frac{N}{d}\right\...
36 views

### On odd perfect numbers and a GCD - Part III

(Note: This post is an offshoot of this earlier MSE question.) In what follows, we let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. We also let $D(x)=2x-\sigma(x)$ denote the ...
19 views

### Why does the divisor-counting function appear in bounds for Kloosterman sums?

Given integers $m,n$ and $c \geq 2$, the Kloosterman sum is defined as $S(m,n;c) = \sum_{k \in (\mathbb{Z}/c\mathbb{Z})^{\times}}{e^{\frac{2i\pi}{c}(mk+nk^{-1})}}$, where $k^{-1}$ is the reciprocal of ...
29 views

### About the characterization of solutions of an equation that involves particular values of the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
20 views

### A total function is representable iff it is weakly representable

The book A Friendly Introduction to Mathematical Logic - 2nd Edition by Christopher C. Leary and Lars Kristiansen gives the following proposition without proof: Proposition 5.3.6. Suppose that $f$ ...
51 views

### What is meant by $\sum_{d \le x}f(d)$ in this equation?

Wikipedia's page (here) on the average order of arithmetic functions gives the following as a means of finding such an order using Dirichlet Series: Define $f$ as an arithmetic function on $n$, and ...
20 views

20 views

106 views

### Dirichlet Series in analytic number theory

I have a question about Abscissa of Convergence of Dirichlet series. The question is ; "Let $\sigma_{1}$ and $\sigma_{2}$ be real numbers with $\sigma_{1} \leq \sigma_{2} \leq \sigma_{1}+1 .$ ...
142 views

### A question about Dirichlet series [closed]

i have the following question given 2 options as i) and ii) Let $f(n)$ be the unique positive real-valued arithmetic function that satisfies $\sum_{d | n} f(d) f(n / d)=1$ for all $n$ . (i.e., $f$ ...
26 views

27 views

### Attempt to get a characterization of even perfect numbers from an equation involving the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(n)$ for integers $n\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
### Is there a number $\mathscr{D}_2 \neq \mathscr{D} = {{3003}^2}\cdot{22021}$ satisfying a certain condition?
(Note: This question is tangentially related to this earlier one.) Let $$\sigma(x) = \sum_{d \mid x}{d}$$ denote the sum of divisors of $x \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural ...