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Questions tagged [arithmetic-functions]

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

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About The Sum of Positive Divisors of $n$

The question says: Find the smallest positive integer $n$ so that $\sigma(x)=n$ has no solution, exactly two solutions, exactly three solutions. I could not come up with a good way to solve this ...
2
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1answer
35 views

What proportion of the positive integers satisfies $\gcd(n^2,\sigma(n^2))>\sigma(n)$, where $\sigma$ is the sum-of-divisors function?

The title says it all. What proportion of the positive integers satisfies $\gcd(n^2,\sigma(n^2))>\sigma(n)$, where $\sigma$ is the sum-of-divisors function? I tried using Sage Cell Server to ...
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2answers
40 views

A proof of $\sum\limits_{d|n} \sigma(d) = n \sum\limits_{d|n} {\tau(d) \over d}$

I'm trying to proof the following statement: Let $n \in \mathbb{Z}$ and the $\sum$ are on the divisors $d$ of $n$. Show that $$\sum\limits_{d|n} \sigma(d) = n \sum\limits_{d|n} {\tau(d) \over d}.$$ ...
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1answer
43 views

A question regarding a paper of Ochem and Rao about the radical of an odd perfect number

Let $\operatorname{rad}(n)$ denote the radical or square-free part of the positive integer $n$, that is, $$\operatorname{rad}(n) = \prod_{p \mid n}{p}$$ where $p$ runs over primes. In the paper ...
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2answers
51 views

How to prove $\sum_{d\mid q}\frac{\mu(d)\log d}{d}=-\frac{\phi(q)}{q}\sum_{p\mid q}\frac{\log p}{p-1}$?

Prove that $$\sum_{d\mid q}\frac{\mu(d)\log d}{d}=-\frac{\phi(q)}{q}\sum_{p\mid q}\frac{\log p}{p-1},$$ where $\mu$ is Möbius function, $\phi$ is Euler's totient function, and $q$ is a positive ...
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0answers
23 views

Möbius inversion across all natural numbers (no divisors)

EDITED TO ACCOMMODATE COMMENTS: I'm trying (self-taught) to understand more about Möbius inversion. Take two arithmetic functions $f$ and $g$ defined by $$g(n)=\sum_{d|n}f(d)$$ (Presumably $d$ ...
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54 views

On the quantity $\sigma(\frac{n^2 \sigma(n^2)}{D(n^2)})$ when $q n^2$ is an odd perfect number with special prime $q$

Denote the sum of the divisors of $x \in \mathbb{N}$ by $\sigma(x)$. Also, denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$. If $m$ is odd and $\sigma(m)=2m$, then $m$ s called an odd perfect ...
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1answer
85 views

If $x$ and $y$ are solitary numbers satisfying $\gcd(x,y)=1$, under what conditions does it follow that $xy$ is also solitary?

Let $\sigma(z)$ denote the sum of divisors of $z \in \mathbb{N}$. Denote the abundancy index of $z$ by $I(z) = \sigma(z)/z$. If the equation $I(z)=I(a)$ has the lone solution $z=a$, then $a$ is said ...
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43 views

Sum of a multiplicative arithmetic function

Prime factorization of $n$ is $\prod p_i^{e_i}$ Then radical of $n$ is defined as $\text{rad}(n)=\prod p_i$ Let $S(N) = \sum_{n=1}^{N}\text{rad}(n)$ I want to calculate $S(N)$ for very large value ...
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0answers
61 views

Could a Fermat prime divide an odd perfect number?

This question is a follow-up to this earlier post. A positive integer $N$ is said to be perfect if $\sigma(N)=2N$, where $\sigma(x)$ is the sum of divisors of $x \in \mathbb{N}$. If $M$ is odd and $\...
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0answers
77 views

Does mathlove's answer imply that $D(n^2) \mid n^2$?

In what follows, set $\sigma(x)$ to be the sum of divisors of $x \in \mathbb{N}$, and let $$D(x) = 2x - \sigma(x)$$ be the deficiency of $x$, and let $$s(x) = \sigma(x) - x$$ be the sum of the aliquot ...
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1answer
40 views

Von Mangoldt function with PARI GP

How to program the Von Mangoldt function: $$\Lambda (n)=\begin{cases}\log p&{\text{if }}n=p^{k}{\text{ for some prime }}p{\text{ and integer }}k\geq 1,\\0&{\text{otherwise.}}\end{cases}$$ ...
3
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1answer
100 views

What is the common (and simplified) value for $D(q^k)D(n^2) = 2s(q^k)s(n^2)$ when $q^k n^2$ is an odd perfect number?

In an answer to an earlier question, it is shown that $$D(2^p - 1)D(2^{p-1}) = 2s(2^p - 1)s(2^{p-1}) = 2^p - 2,$$ if $2^{p-1}(2^p - 1)$ is an even perfect number, $D(x) = 2x - \sigma(x)$ is the ...
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2answers
80 views

Is $(q^k n^2 \text{ is perfect }) \iff (D(q^k)D(n^2) = 2s(q^k)s(n^2))$ only true for odd perfect numbers $q^k n^2$?

(Preamble: This question is an offshoot of this earlier MSE post.) The title says it all. Is $\bigg(q^k n^2 \text{ is perfect }\bigg) \iff \bigg(D(q^k)D(n^2) = 2s(q^k)s(n^2)\bigg)$ only true for ...
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1answer
35 views

Can you explain the following property for arithmetic functions.

I have trouble understanding the following identity $$\prod_{d|2n} (x^d-1)^{\mu (2n/d)} = \prod_{d|n} (x^d-1)^{\mu (2n/d)} \prod_{d|n} (x^{2d}-1)^{\mu ((2n/d)/2)}$$, $\mu (n)$ is the Mobius function. ...
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0answers
51 views

Are there any other squares $n^2$ for which $\gcd(n^2, \sigma(n^2)) = 2n^2 - \sigma(n^2)$?

Let $\sigma(x)$ denote the sum of the divisors of the positive integer $x$. Denote the deficiency of $x$ by $$D(x)=2x-\sigma(x).$$ I am interested in solutions to the equation $$\gcd(n^2, \sigma(n^2)...
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1answer
73 views

Number of pairs $(a,b)$ such that $a+b<n$ and such that $\gcd(a,b,n)=1$

Let $n$ be a positive integer. What is the number of pairs $(a,b)$ of positive integers such that $a+b<n$ and such that $\gcd(a,b,n)=1$? I know that the number of positive integer $a$ such that $\...
1
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1answer
43 views

If an arithmetic function $f$ is such that $\sum\limits_{n=1}^Nf(n)=\Theta(N)$ then $f(n)=o(n^\epsilon)$

Consider a positive-valued, arithmetic function $f$ with $f(n)\geq 2$. Suppose that $f$ satisfies the inequality $$c_1N\leq\sum_{n=1}^Nf(n)\leq c_2N$$ where $0<c_1<c_2$ are real constants. ...
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0answers
41 views

Help proving an equality about arithmetic functions

So I have this equality to prove for any $n \in \mathbb{N}$: $$ \sum_{d|n} \sigma(d) \phi(\frac{n}{d}) = n \tau(n) $$ So I was able to show that left and right side are multiplicative. So how can I ...
4
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1answer
282 views

Is it possible to simplify this expression even further?

(Preamble: This question is tangentially related to this earlier one.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $...
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0answers
262 views

A conjecture regarding odd perfect numbers

(Note: This question has now been cross-posted to MO.) Let $\sigma(z)$ denote the sum of the divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the deficiency of $z$ by $D(z):=2z-\...
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3answers
56 views

Is there a lower bound for $\operatorname{rad}(n)$ in terms of $n \in \mathbb{N}$?

For integers $n\geq 1$ we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an integer $n>1$ ...
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4answers
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Do odd numbers $n$ satisfying $\gcd(n, \sigma(n)) > \sqrt{n}$ have a special form?

Let $\sigma(n)$ denote the sum of divisors of the positive integer $n$. Using Sage Cell Server, I was able to get the following odd numbers $n < 5000$ satisfying $\gcd(n, \sigma(n)) > \sqrt{n}$:...
4
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1answer
64 views

A bound for a sum over square-free numbers: $\sum_{n \leq X} \frac{\mu(n)^2 \tau_k(n)}{\phi(n)} \ll (\log X)^k$

How can one show that $$\sum_{n \leq X} \frac{\mu(n)^2 \tau_k(n)}{\phi(n)} \ll (\log X)^k \ ?$$ Here, $$\tau_k(n) = \sum_{\substack{d_1,d_2, \dots, d_k \in \mathbb{N} \\ d_1d_2\cdots d_k=n}} 1.$$ I'm ...
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0answers
31 views

Does $D(n)$ always depend on $\gcd(n,\sigma(n))$ when $\sigma(N)=aN+b$, $N=xn$, and $n$ is a square?

Does $D(n)$ always depend on $\gcd(n,\sigma(n))$ when $\sigma(N)=aN+b$, $N=xn$, and $n$ is a square (with $\gcd(x,n)=1$)? Here, $\sigma$ is the sum of divisors and $D(n) := 2n - \sigma(n)$ is the ...
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0answers
18 views

Deduce the number of divisors, $d(n)$ is multiplicative and obtain a formula of $d(n)$ in terms of prime decomposition [duplicate]

I have been given a question which begins with me deducing $d(n)$ is multiplicative, I know $d(n)= \sum_{d|n} 1,$ and obtain a formula of $d(n)$ in terms of prime decomposition (once again I know is $...
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0answers
16 views

Inverting arithmetic functions

I know that if $$F(n)=\sum_{d|n}f(d)$$ then $$f(p^e)=F(p^e)-F(p^{e-1})$$ Let $F(n)=\sum_{d|n}f(d)$ and assume $F(n)$ is multiplicative, so $F(1)=1$. Find the formula for $f(n)$ when $F(n)=1$ for all $...
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1answer
22 views

Inverse of a Multiplicative Arithmetic Function w/0 Mobius function

So I was given that if $$F(n)=\sum_{d|n}f(d)$$ then $$f(p^e)=F(p^e)-F(p^{e-1})$$ I'm trying to understand how to use this to actually solve specific problems. For example, let $F(n)=\sum_{d|n}f(d)$ ...
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2answers
52 views

If $f$ is an arithmetic function with $f(1)=1$ then $f$ is multiplicative

I'm studying analytic number theory for undergraduates and I read this theorem in Tom Apostol's book on the second chapter: Theorem 2.12. If $f$ is multiplicative then $f(1)=1$ And under need ...
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1answer
25 views

Is the average order of a product of arithmetic functions the product of the average orders?

If we have $$\sum_{n \leq x} f(n) \sim \sum_{n \leq x} g(n)$$ $$\sum_{n \leq x} h(n) \sim \sum_{n \leq x} k(n)$$ does it follow that $$\sum_{n \leq x} f(n)h(n) \sim \sum_{n \leq x} g(n)k(n)$$ In ...
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1answer
151 views

Is it true that $\underbrace{x^{x^{x^{.^{.^{.^x}}}}}}_{k\,\text{times}}\pmod9$ has period $18$ and can never take the values $3$ and $6$?

Is it true that the arithmetical function $f:\mathbb{N}\setminus\{1\}\rightarrow \mathbb{Z}_9$ given by $$f(x\mid k)=\underbrace{x^{x^{x^{.^{.^{.^x}}}}}}_{k\,\text{times}}\pmod9$$ has period $...
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0answers
27 views

Is feasible a simple inequality as a combination of the Firoozbakht's conjecture and the properties of the Ramanujan tau function?

I would like to know if is feasible a nice/potentially interesting combination of the Firoozbakht's conjecture, this Wikipedia, and the Ramanujan tau function using the so-called Ramanujan conjectures ...
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1answer
37 views

Multiplicative arithmetic function on the unit disk

Suppose $f$ is a multiplicative arithmetic function that takes values inside the unit disk, and let Re$(s)>1$. We define $F(s) = \sum_{n\ge1}^{}\dfrac{f(n)}{n^s}$. I want to show that $$\text{log } ...
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1answer
63 views

$Var[w]=O(\log\log(N))$ where $w(n)$ is the no of prime divisors of $n$

I was seeing the question Here So, I have a doubt at the last line of the particular question. Let me frame in the same way wolf has done it, the main objective is to prove the same thing of Marius ...
0
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2answers
78 views

Prove $ \left\vert \sum_{n=1}^N \frac{\mu(n)}{n} \right\vert \leqslant 1,$ where $\mu(n)$ is the Mobius function.

Given a positive integer $N$, show that $$ \left\vert \sum_{n=1}^N \frac{\mu(n)}{n} \right\vert \leqslant 1,$$ where $\mu(n)$ is the Mobius function. How do I approach this question? I guess a ...
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0answers
31 views

Proving arithmetics results using Jacobi Identity

I am taking a course on Complex Analysis and we have to solve the following problem: Given the identity $F(z^4)=(G(z))^4$ for $F(z)=\sum_{k=0}^\infty \frac{(2k+1)(z^{2k+1})}{1-z^{4k+2}}$ and $G(z)=\...
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0answers
48 views

Prove the asymptotic probability of numbers being relative prime equals $\frac{6}{\pi^2}$

For a number theory problem, I'm trying to prove that the asymptotic probability of numbers being relative prime equals $\frac{6}{\pi^2}$. Unfortunately, I'm a bit stuck at the following equality. $$\...
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0answers
22 views

Variations of Somos's recurrence for number theoretic functions, with some special and similar property

While I was reading articles from MathWorld I've found the article dedicated to the so-called Somos's Quadratic Recurrence Constant. I would like to know if it is possible to define some (a different)...
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2answers
58 views

Equivalent formula for the sum of first $n$ values of the number of divisors function

In the notes of the following OEIS sequence( https://oeis.org/A006218), it is stated that $$\sigma_0(1) + \sigma_0(2) +... + \sigma_0(n) = \left[ \dfrac{n}{1} \right] + \left[ \dfrac{n}{2} \...
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0answers
26 views

On mobius and Dirichlet type transformation of complex valued functions on $[1,\infty )$

For $f : [1,\infty) \to \mathbb C$, it is known that $g(x)= \sum_{1\le n \le x, n \in \mathbb N} f(x/n), \forall x \in [1,\infty)$ iff $f(x)=\sum_{1\le n \le x, n \in \mathbb N} \mu(n) g(x/n), \forall ...
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votes
1answer
44 views

Write the equation $\sin^2(x)-3/4$ in alternate form

The equation is: $$\sin^2(x)-(3/4)$$ How do you get this equation in an alternate form, so that there is only division or multiplication present between trigonometric functions. The solution is: $$\...
1
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1answer
39 views

Arithmetic function sums

So there in alternative proof illustrated in my text it goes: $$\sum_{a=1,(a,n)=1}^n e^{2\pi ia/n} = \sum_{a=1}^n e^{\frac{2\pi ia}{n}}\sum_{d|(a,n)}\mu(d)---(1)$$ $$=\sum_{a=1}^n\sum_{d|(a,n)} e^{\...
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0answers
25 views

On a notion of “period” of Dirichlet character, modulo $k$, restricted to integers co-prime to $k$

Let $\chi : \mathbb Z \to \mathbb C$ be a Dirichlet character mod $k$ ; i.e. $\chi (m+k)=\chi (m) , \forall m \in \mathbb Z$ ; $\chi (mn)=\chi(m)\chi (n),\forall m,n \in \mathbb Z$ and $\chi (n)=0$ ...
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votes
1answer
46 views

Find all functions $f:\Bbb N\rightarrow \Bbb R $ such that: $f(m+k)=f(mk-n)$ [duplicate]

Find all functions $f:\Bbb N\rightarrow \Bbb R $ such that for a given value $n\in \Bbb N$ , the following identity holds: $$f(m+k)=f(mk-n) ,m,k \in \Bbb N , mk>n$$ This problem has already ...
2
votes
2answers
69 views

Arithmetic Functions Summation

Ok so this was one of the problems in my book, and it goes: Show that for each positive integer $n$:$$\sum_{a=1,(a,n)=1}^n e^{\frac{2\pi a i}{n}} = \mu(n)$$. $Proof$ : Let $\Phi(n) = \sum_{a=1,(a,n)=...
4
votes
4answers
83 views

What does $(-1)^{2/3}$ equal?

I figured that according to the exponent laws it should equal $1$ since: $$(-1)^{2/3} = ((-1)^2)^{1/3} = 1^{1/3} = 1$$ But according to wolframalpha and google it equals something imaginary. Why is ...
2
votes
0answers
29 views

Compare $\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\rfloor\right)}$ and $\sum_{k\mid n}k^{\operatorname{rad}\left(\frac{n}{k}\right)}$

I would like to know how do a comparison between the sizes of these functions defined for integers $n\geq 1$, when $n$ is large $$f(n):=\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\...
1
vote
1answer
60 views

Show that the quotient $\frac{\sigma(p^3)}{\tau(p^3)}$ is an integer for $p$ prime

Let p $\geq$ 3 be a prime. Show that the quotient $\frac{\sigma(p^3)}{\tau(p^3)}$ is an integer. I know that I have to use the product formulas but not exactly sure how to go from there.
2
votes
0answers
51 views

Upper and lower density of the set of natural numbers whose sum of positive divisors is a perfect square

Let $A:=\{n \in \mathbb N : \sigma(n) \text{ is a perfect square}\}$ . I can show that $A$ is infinite . Let $A(n):=|A \cap [1,n]|$ . What can we say about $\liminf A(n)/n$ and $\limsup A(n)/n$ ?
2
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0answers
50 views

On Uniform Elementary Estimates of Arithmetic Sums Error Term

Stefan A. Burr's paper "On Uniform Elementary Estimates of Arithmetic Sums" has this result: Suppose $G(s)=\sum_{n=1}^{\infty}\frac{g(n)}{n^s},$ $G_2(s)=|g(1)|+\sum_{n=2}^{\infty}\frac{|g(n)|\...