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

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19
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2answers
158 views
+50

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there ...
-3
votes
0answers
20 views

what is the sales price added to the sales tax? [on hold]

The discount on a new computer was $140. This was a discount of %16. What would a customer pay for the computer if a 4% sales tax was added to the sales price. and show work
7
votes
2answers
122 views

Arithmetic Derivatives: Arithmetic Logarithmic Derivative Problem

In Calculus, whenever we see a constant and want to take the derivative of it, it always is 0. However in Number Theory, we have something called the arithmetic derivative in which we can ...
4
votes
1answer
86 views

Infinite exponentiation $n^{n^{n^{…^n}}} \equiv m \pmod q$ , find m?

let $(n,q) \in \mathbb N^{*^2}$ I was wondering if it was possible to find a function $f_q$ such that : $f_q(n)=m$ where $m$ is such that $n^{n^{...^n}} \equiv m \mod q$ or at least an easy way to ...
2
votes
0answers
20 views

Characters appearing naturally in arithmetic functions

Let $r_2(n)$ denote the number of representations of $n$ as a sum of 2 squares. It is well-known that $$r_2(n) = 4\sum_{d \mid n} \chi(d),$$ where $\chi$ is the non-principal character modulo $4$. ...
-1
votes
2answers
45 views

Formula to round decimal values

I'm using an application, which offers a feature of creating user-defined functions. Available set of methematical operations which could be incorporated is rather small, namely I can use: addition ...
0
votes
0answers
33 views

Modified arithmetical functions from a modified Möbius function

In this post when $n>1$ we assume that $n=\prod_{k=1}^{\omega(n)}p_{n}^{e_{k}}$ its factorization in prime powers, where $\omega (n)$ is the number of distinct primes. It is well known the ...
0
votes
1answer
27 views

Composition of analytic function with arithmetic function

Consider an arithmetic function $g$ with codomain $\{a,b\}$ and a function $f$ which is analytic on some domain including $\{a,b\}$. We therefore have $$f(g(n))=\sum_{k=0}^\infty c_k (g(n)-a)^k$$ and ...
3
votes
3answers
185 views

Ramanujan's tau function identity

While studying Ramanujan's tau function, I observed that the function satisfies a beautiful identity that I had not seen previously in the literature. Let $\tau(n)$ be Ramanujan's tau function, such ...
2
votes
1answer
44 views

Closed-Form Modular Arithmetic

Is there a way to define modulo division (or functions of modular arithmetic in general) as superposition of (elementary?) functions? For example, the multiplication is first introduced as ...
7
votes
0answers
129 views

Is the Euler function $\phi$ constant in arbitrarily large intervals?

Is it true that for every $k \in \mathbb{N}$ there exists a natural number $x$ such that $\phi(x)=\phi(x+1)=\cdots=\phi(x+k)$, where $\phi$ is the Euler's totient function? I thought about this ...
1
vote
1answer
39 views

For all $n\in\Bbb N$ does there exist $n_1$ and $n_2$ such that $d(n_1) +d(n_2)=n$

I am trying to show that for each positive integer, $n$, we can find integers $n_1,n_2$ that satisfy $$d(n_1) +d(n_2)=n$$ where $d(n)$ is the divisor function. I am frustrated because even with easy ...
1
vote
0answers
66 views

What's is the name of this function?

A function, $f:\mathbb{N}\to\mathbb{N}$, is defined in the following way, \begin{equation} f(n)=\#\{m\mid m\leq n\text{ and there does not exists any integer }m'>m\text{ such that }m\text{ divides ...
0
votes
1answer
24 views

If an arithmetic function has a strong logarithmic mean value, then it has an ordinary mean value.

Say that an arithmetic function f has a strong logarithmic mean value A, and write L(f) = A, if f satis es an estimate of the form \begin{align} \sum\limits_{n=1}^{x}\frac{f(n)}{n} = Alog(x) + B + ...
1
vote
0answers
36 views

Bezout Coefficient for polynomials in sage?

I want to find the bezout coefficient for those 2 polynomials : $f = 1+x-x^2-x^4+x^5$ and $g = -1+x^2+x^3-x^6$ when I use the gcd function in sage the output is : ...
2
votes
1answer
40 views

for which values of the pair of integers $(n,k)$ is $p(n,k) =1+\frac{2^{k}-1}n$ is prime?

let $p(n,k)= 1+\frac{2^{k}-1}{n}$ for a positive integer $n,k$ -for which values of the pair of integers $(n,k)$ : $p(n,k)$ is prime ? Any help is very welcom .Thank you
5
votes
0answers
170 views

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
3
votes
0answers
76 views

Approximate how the Numbers $n$ such that Mertens' function is zero grow.

Is it possible to approximate how the "Numbers $n$ such that Mertens' function is zero" grow?
4
votes
0answers
331 views

Numbers $n$ such that Mertens' function is zero.

OEIS (A028442) lists the Numbers n such that Mertens' function $$ M(n)=\sum_{k=1}^n\mu(k) $$ is zero: 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, ...
4
votes
1answer
45 views

Number of Divisors of most numbers

In the book A Comprehensive Course in Number Theory by Alan Baker. The author mentions that even though the average order of $\tau(n)$ is $\log n$, almost all numbers have about $(\log n)^{\log 2}$ ...
3
votes
1answer
58 views

Convolution identity involving the Möbius function $\sum_{d|n,d>0} |\mu(d)| = 2^{\omega(n)}$

I'm learning about the Möbius Inversion Formula but I'm stuck on an exercise which involves the Möbius function. Let $n\in\mathbb{Z}$ with $n>0$ and let $\omega(n)$ denote the number of distinct ...
0
votes
1answer
25 views

Multiplicative Function Property

Suppose $f$ is an arithmetic function. Does the property $f(1) = 1$ if $f$ is multiplicative? Or is it only when $f$ is completely multiplicative?
5
votes
1answer
84 views

A question about Euler's totient function

Prove that for every natural number $m$, there exists a natural number $n$ such that $$\phi(n)=\phi(n+m)$$ For odd numbers $m$, we can choose $n=m$ and use the identity $\phi(2m)=\phi(m)$.
3
votes
5answers
129 views

How can I show that $\phi(m) \mid \phi(n)$? [duplicate]

I want to prove that: $$\text{ if } m,n \geq 1 \text{ and } m \mid n,\text{ then } \phi(m) \mid \phi(n).$$ How can I show this? I thought the following: $$m \mid n \Rightarrow \exists k \in ...
2
votes
2answers
141 views

Proving $\phi(m)|\phi(n)$ whenever $m|n$ [duplicate]

Show that $\varphi(m)|\varphi(n) $ whenever $m|n$. I am stuck after writing the formula. I know that if $m$ divides $n$, that means one of the prime factors of $n$ would include $m$ or a multiple of ...
3
votes
0answers
131 views

To prove $\phi(mn)\phi(d)=\phi(m)\phi(n)d$ without explicitly computing the phi function values

If $m,n$ are positive integers with g.c.d.$(m,n)=d$ , then we can show by explicitly computing respective totients that $\phi(mn)\phi(d)=\phi(m)\phi(n)d$, I want to know, is there any more elegant way ...
3
votes
1answer
84 views

find all positive integers n such that $\phi(n)+\sigma(n)=2n$.

I'm asked to find all integers n such that $\phi(n)+\sigma(n)=2n.$ I know that when n is a prime, $\phi(n)+\sigma(n)=(n-1)+(n+1)=2n.$ My guess is that n can only be primes, and I want to derive a ...
0
votes
1answer
20 views

On the sets $\{\dfrac {\phi(n+1)}{\phi(n)} : n\in \mathbb Z^+\}$ and $\{\dfrac {\phi(n)}{n} : n\in \mathbb Z^+\}$

How to show that the set $\{\dfrac {\phi(n+1)}{\phi(n)} : n\in \mathbb Z^+\}$ is dense in $\mathbb R^+$ ? Also that the set $\{\dfrac {\phi(n)}{n} : n\in \mathbb Z^+\}$ is dense in $(0,1)$ ?
2
votes
0answers
53 views

how to show $\lim \sup_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= \infty$ and $\lim \inf_{n \to \infty} \frac{\phi(n+1)}{\phi(n)}= 0$?

Let $\phi (n)$ be the Euler's totient function . Thenhow to prove that the set $\{\dfrac{\phi(n+1)}{\phi(n)}: n \in \mathbb Z^+\}$ is unbounded above that is how to show $\lim \sup_{n \to \infty} ...
1
vote
3answers
63 views

The probability that a random integer is prime to a given integer $m$ is $\frac{\phi(m)}m$?

Let $N_m(x)$ denote the number of positive integers not exceeding $x$ that are relatively prime to $m$ , then how to prove that $\lim_{x \to \infty} \dfrac{N_m(x)}x=\dfrac{\phi (m)}m$ , where ...
4
votes
1answer
105 views

Prove $\sum_{k = 1}^n \mu(k)\left[ \frac nk \right] = 1$ [duplicate]

I need to prove the identity $$\sum_{k = 1}^n \mu(k)\left[ \frac{n}{k} \right] = 1$$ where $n$ is a natural number, and $[n]$ denotes the floor function. The proof also should not use the Möbius ...
6
votes
2answers
811 views

Prove $\sum_{d \leq x} \mu(d)\left\lfloor \frac xd \right\rfloor = 1 $

I am trying to show $$\sum_{d \leq x} \mu(d)\left\lfloor \frac{x}{d} \right\rfloor = 1 \;\;\;\; \forall \; x \in \mathbb{R}, \; x \geq 1 $$ I know that the sum over the divisors $d$ of $n$ is zero if ...
1
vote
1answer
75 views

Prove $\lambda(n)=\sum_{d^2|n}\mu(n/d)^2$ and $\mu^2(n)=\sum_{d^2|n}\mu(d)$

$\lambda(n)$= $\sum_{d^2|n}$ $\mu(n/d)^2$ and $\mu^2(n)$= $\sum_{d^2|n}$ $\mu(d)$ Having a little bit of trouble here.Can I use the fact that $\sum_{d|n}\lambda(n)$ is a characteristic function for ...
0
votes
1answer
33 views

Proof involving the Euler phi-function

I'm having trouble approaching the following exercise: Let $n$ and $k$ be positive integers. Prove that $\phi(n^k) = n^{k-1}\phi(n)$. I've tried examining the prime factorization $n^k = (p_1^{a_1} ...
7
votes
1answer
124 views

Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised ...
1
vote
0answers
59 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetic functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
2
votes
2answers
497 views

Show that $\sigma(n) = \sum_{d|n} \phi(n) d(\frac{n}{d})$

This is a homework question and I am to show that $$\sigma(n) = \sum_{d|n} \phi(n) d\left(\frac{n}{d}\right)$$ where $\sigma(n) = \sum_{d|n}d$, $d(n) = \sum_{d|n} 1 $ and $\phi$ is the Euler Phi ...
3
votes
1answer
73 views

Formula for $\sum_{d|n} \frac {\mu(d)}d$

I feel like I've seen a formula somewhere for $\displaystyle \sum_{d|n} \frac {\mu(d)}d$, but I can't remember what it is and can't find it. Does anybody know of a formula?
8
votes
3answers
950 views

Is there a “nice” formula for $\sum_{d|n}\mu(d)\phi(d)$?

I'm trying to work through Ireland and Rosen's A Classical Introduction to Modern Number Theory as I've heard good things about it. This is Exercise 12 from Chapter 2. Here $\mu$ is the Moebius ...
1
vote
2answers
132 views

A proof of $\sum{\mu(n)/n}=0$

I am looking for a proof (or references) of the following statement $$\sum_{n=1}^{\infty}{\frac{\mu(n)}{n}}=0$$ where $\mu$ is the Möbius function. Many thanks !
1
vote
1answer
284 views

Question - Möbius inversion formula

I need your help in the next question: Prove directly from the definition the Möbius inversion formula. (Möbius function defined as follows: μ(n) = 1 if n is a square-free positive integer with ...
10
votes
5answers
529 views

Identity with nested sum taken over divisors of $\gcd$'s

For computational reasons, I want to show that the following holds true: Let $n_1,n_2,N\in \mathbb{N}$. One has $$\Large \sum_{a\mid \gcd(n_1,N)}\sum_{b\mid \gcd(n_2,\frac{n_1N}{a^2})} ab ...
0
votes
0answers
38 views

Order of operations for log transformation

I am working with a large dataset of positive values with a positive skew. I will be using a Ln transformation in SPSS to normalize my dataset. However, I am not sure of the order of operations. For ...
1
vote
1answer
45 views

Demonstration in arithmetic function

I need help to know (in detail) how to prove that the product of two multiplicative arithmetic functions is a multiplicative arithmetic function. $$$$$f(n)$ and $g(n)$ are functions multiplicative, ...
-1
votes
5answers
165 views

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$ I found (by trial) $m=\{13,21,26,28,36\}$, but do not know if misinterpreted the problem, but actually I suppose I have to find an equation ...
1
vote
0answers
64 views

Summing the reciprocal of $\phi(n)^2$

I am currently reading Vaughan's book 'The Hardy-Littlewood method' and am working on exercise 3.3 (page 37). I have tried working through a special case but that didn't help either. More concretely, ...
6
votes
1answer
88 views

Lower bound of Euler phi function times sum of divisors

After some work, I got this nice inequality: $$ \frac{n^2}{2} < \phi(n)\cdot \sigma(n) $$ where $\phi(n)$ is Euler's phi function and $\sigma(n)= \sum_{d|n} d$. I know this is true because I'm ...
4
votes
2answers
79 views

Proving an identity of the Möbius function and Euler’s totient function product

Could anyone kindly help me to prove that $$ \sum_{d|n} \mu(d) \varphi(d) = 0 $$ for all even integers $ n \geq 2 $, where $ \mu $ is the Möbius function and $ \varphi $ is Euler’s totient function? ...
0
votes
2answers
16 views

Offset a range of numbers

I want to find an equation to offset a range of numbers by a given amount. I'm not sure if I am using the term offset correctly. Lets say the range is from 0 - 1 and I want it to be offset by .25 : 0 ...
2
votes
0answers
24 views

To calculate what is $\sum_{1 \le m<n ;(m,n)=1} m^2$ or what is the remainder when $\sum_{1 \le m<n ;(m,n)=1} m^2$ is divided by $n$? [duplicate]

For an integer $n >1$ , what is the sum of the squares of all the positive integers that are less than $n$ and relatively prime to $n$ that is I am trying to calculate $f(n):=\sum_{1 \le m<n ...