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

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3
votes
1answer
54 views

Hints to compute if exists $\lim_{n\to\infty}\sum_{k=1}^n\sigma(k^2)/\sum_{k=1}^n\sigma(k)$, which $\sigma(n)=\sum_{d\mid n}d$, and other question

I would like receive hints at least for one of the following problems, these are going from experiments. Can you provide to me hints for at least one of the following problems? I will try put the ...
1
vote
0answers
43 views

Sketch of a possible equivalence with Riemann hypothesis

From Robin's equivalence (see [1]) and the following trigonometrics identitites, I ask to me if it is feasible write vagues equivalences using this strong result and if these equivalences will be ...
4
votes
1answer
48 views

On a generalization for $\sum_{d|n}rad(d)\phi(\frac{n}{d})$ and related questions

Let $\phi(m)$ Euler's totient function and $rad(m)$ the multiplicative function defined by $rad(1)=1$ and, for integers $m>1$ by $rad(m)=\prod_{p\mid m}p$ the product of distinct primes dividing $...
1
vote
1answer
41 views

Arithmetic functions & logs

How do you simplify the following equation in terms of other arithmetic functions? $$ f(n)= \sum_{d|n} \mu(d) log(d) ? $$ log(n) is not a multiplicative arithmetic function, so i dont know what to ...
1
vote
2answers
39 views

With $s(n)=\sum_{k=1}^n n \bmod k$, can be justified that $\forall\epsilon>0$ let us $\lim_{n\to\infty}\frac{s(n-1)}{\epsilon+s(n)}=1?$

Denoting as $$s(n)=\sum_{k=1}^n n \bmod k$$ the sum of remainders function (each remainder is defined as in the euclidean division of integers $n\geq 1$ and $k$). See [1] for example. For examples $...
10
votes
3answers
356 views

Arithmetical Functions Sum, $\sum_{d|n}\sigma(d)\phi(\frac{n}{d})$ and $\sum_{d|n}\tau(d)\phi(\frac{n}{d})$

$$\sum_{d|n}\sigma(d)\phi\left(\frac{n}{d}\right)=n\tau(n) ,\\ \sum_{d|n}\tau(d)\phi\left(\frac{n}{d}\right)=\sigma(n)$$ The problem (7.4.15) of Burton's Elementary Number Theory has been request to ...
1
vote
1answer
45 views

Using Dirichlet convolution where f = μ ∗ μ (Mobius) to find f(24)?

I am confused about the Dirichlet convolution and how it is used. Does it take two entirely different arithmetic functions? And knowing that f = μ ∗ μ (the Mobius function), why does the question I ...
2
votes
1answer
71 views

Infinite sum of a function $g(n)=\sum_{d|n \; d\ne n}g(d)$

Let the function $g:\Bbb{N}\to\Bbb{N}$ be defined as $$g(n)=\sum_{d|n \; d\ne n}g(d)$$ with $g(1)=1$, how can we evaluate a sum like $$\sum_{i=0}^\infty{g(15^i)\over15^i} \tag1$$ Need we find a ...
0
votes
0answers
45 views

Show that if $n$ is even then $\sum_{d|n}\mu(d)\phi(d)=0$ using only Dirichlet Convolution propertys (without multiplicative function concepts)

Show that $\displaystyle\sum_{d|n}\mu(d)\phi(d)=0$ using only Dirichlet Convolution propertys (without multiplicative function concepts). I suspect you have to use that $1\ast \mu=I$ and $f\ast 1=id$...
4
votes
2answers
128 views

Was this arithmetic Möbius/Mangoldt function ever used for something?

Definition: Let $n=\prod_k p_k^{c_k}$, with $p_k \in \mathbb P$ and $$ A(n)=\sum_{d|n} \mu(d)\Lambda(d)=\sum_\limits{c_k\neq 0} \log p_k , $$ with the Möbius function $\mu(n)$, which is: $...
1
vote
0answers
37 views

Numbers $n$ whose prime factors are $2$ and $5$ if and only if $\sum_{k=1}^{rad(n)}\mu(k)k=0$?

In the literature if defined the arithmetical function $rad(n)$ as $1$ if $n=1$ and for $n>1$ by $rad(n)=\prod_{p|n}p$. It is obviously a multiplicative function. Too we know the Mobius function $\...
2
votes
2answers
28 views

On $\sigma(n)<n+\frac{n}{2}\log\frac{n+1}{n-1}+\frac{n}{4} \left(\sum_{\substack{d|n,1<d<n}}\log\frac{(n+d)(n+1)}{(n-d)(n-1)}\right)$

I've derived for $n>1$ and $\sigma(n)$ the sum of divisor function $\sum_{d|n}d$ the following inequality $$\sigma(n)<n+\frac{n}{2}\log\frac{n+1}{n-1}+\frac{n}{4} \left(\sum_{\substack{d|n,1<...
3
votes
1answer
39 views

The sum $\sum_{n\leq x}\sum_{\substack{1\leq k\leq n \\ gdc(k,n)=1}}cos^2\pi \frac{k}{n}$ diverges as $x$, when $x$ tends to infitity

I want to know if it is possible find an easy proof (this is without an use of an strong result) of Question. Prove that the following sum diverges as $x\to\infty$ $$\sum_{n\leq x}\sum_{\...
0
votes
0answers
31 views

On $\sum_{k\nmid n}k$, where the sum is over the integers $1\leq k\leq n$ such that $k\nmid n$, and perfect numbers

If we define the arithmetic function $\delta(n)$ as the sum of integers $1\leq k\leq n$ such that $k\nmid n$, we have by Gauss statement $\sum_{k=1}^n k=n(n+1)/2$, that $$\sigma(n)+\delta(n)=\frac{n(...
0
votes
2answers
57 views

Formula with $\sigma(n)$ and $\phi(n)$ is not an integer if $n$ is not square-free

The question Let $n$ be an integer, let $\sigma(n)$ be the sum-of-divisors function, and let $\phi(n)$ be Euler's totient function. Prove that $\frac{\phi(n)\sigma(n) + 1}{n}$ is an integer if $n$ ...
0
votes
0answers
41 views

Partial GCD - Sum

$\sum\limits_{n = 2}^{M} \sum\limits_{m = 1}^{R} GCD(m,n) $ $R = (\lfloor\dfrac{N}{n}\rfloor-n) \% n $ $M = \lfloor\sqrt{N}\rfloor $ I calculated that - For N=10 the sum is 1 For N = 100 the ...
4
votes
0answers
140 views

Arithmetic Derivative

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 ...
2
votes
1answer
34 views

On Abel summation $\sum_{e<n\leq x}\left(\mu(n)\cdot\int_2^n\frac{ds}{\log s}\right)$, where $\mu(n)$ is the Möbius function

By Abel's identity for $Li (x)=\int_2^x\frac{ds}{\log s}$, $a(n)=\mu(n)$ the Möbius function and $[y=e,x]$ (see Theorem 4.2, page 77 of [1]) and an application of Fundamental Calculus Theorem we ...
0
votes
1answer
33 views

Number of zeros of a polynomial modulo n is a multiplicative function

Let $f$ be a polynomial with integer coeffcients. For $n\geq1$ let $N_f(n)$ denote the number of pairwise incongruent solutions of $f(x)=0$ mod n. I need help proving that $f$ is a multiplicative ...
4
votes
0answers
331 views

LCM of binomial coefficients and related functions

I know about the following identity: $$\displaystyle \text{lcm} \left( {n \choose 0}, {n \choose 1}, ... {n \choose n} \right) = \frac{\text{lcm}(1, 2, ... n+1)}{n+1}$$ 1) Is there any method to find $...
3
votes
1answer
76 views

Looking an asymptotic for $\sum_{k\leq x}\Lambda(k)e^k$, where $\Lambda (n)$ is the von Mangoldt function

Using Abel's identity (see Theorem 4.2 in page 77 of [1]) and Prime Number Theorem (Theorem 4.4 in page 75) I compute $$\frac{1}{x}\sum_{k\leq x}\Lambda(k)e^k\sim 1\cdot e^x-\frac{1}{x}\int_1^x \psi(...
4
votes
0answers
77 views

Derivatives, discrete and continuous, of $(1/\sqrt{n})\cos (t\log n)$ and $(1/\sqrt{n})\sin (t\log n)$ and Cauchy-Riemann equations

For any arithmetical function $f(n)$, we define its derivative to be $f'(n)=f(n)\cdot \log n$ for $n\geq 1$ (see for example [1], page 45 or Wikipedia). Fact. The functions $u(n,t)=(1/\sqrt{n})\...
0
votes
0answers
28 views

On $\sum_{d|n}(\mu(n/d)\sigma_0(d)/2)\log d$, where $\sigma_0(n)$ and $\mu(n)$ are, respectively, the number of divisors and Möbius functions

Let $f(n)=\sum_{d\mid n}\log d$. Then using Möbius inversion formula (see Wikipedia) and $\prod_{d\mid n}d=n^{\sigma_{0}/2}$ (see [1], page 47) where $\sigma_{0}(n)$ is the number of positive divisors ...
1
vote
0answers
45 views

What happens to the $\color{red}s$ in Möbius' Inversion Formula?

At the end of the Wiki page on Möbius' Inversion Formula, the following relation is given: $$ g(x) = \sum_{m=1}^\infty \frac{f(mx)}{m^\color{red}s}\quad\mbox{ for all } x\ge 1\quad\...
19
votes
2answers
198 views

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 ...
6
votes
2answers
217 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
114 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
22 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
66 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 ...
1
vote
0answers
55 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
29 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
307 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
76 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 summation,...
7
votes
0answers
155 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
45 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
70 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
30 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 + o(...
1
vote
0answers
80 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
45 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
180 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
86 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
366 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
49 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
115 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
28 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
90 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
155 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
189 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 ...
4
votes
0answers
203 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
209 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 ...