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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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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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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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31 views

proof that $ \tau(n^a)=\sum_{d|n} a^{w(d)} $

$$ \tau \ is \ the\ number\ of \ positive \ divisors , and\ w(n) \ the \ number\ of \ distinct \ primes \ of \ n \ and\ a\in\mathbb{N} $$ $$ If\ it\ holds\ for\ p^m then \ the \ equation\ is \ ...
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1answer
30 views

Von Mangoldt function with PARI GP

How to program the Von Mangoldt function: $$\Lambda (n)=\begin{cases}\log p&{\text{if }}n=p^{k}{\text{ for some prime }}p{\text{ and integer }}k\geq 1,\\0&{\text{otherwise.}}\end{cases}$$ ...
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1answer
79 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
71 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
32 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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45 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 $\...
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1answer
40 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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39 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 ...
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1answer
280 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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255 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
47 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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1answer
47 views

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 $\gcd(n, \sigma(n)) > \sqrt{n}$:...
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1answer
62 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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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
17 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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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
21 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
49 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
24 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
149 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
36 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
57 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 ...
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2answers
71 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
28 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
47 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
21 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
54 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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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: $$\...
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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
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Formula for $\phi(x/y) = \#\{ a/b \in \Bbb{Q}_{\gt 0} : (a,b) = (ab, xy) = 1, 1 \leq a \leq x, 1 \leq b \leq y\}$?

Suppose we define two rationals $x/y, \ a/b$ to be coprime when $(ab, xy) = 1$. Then what is an extension of $\phi$, the Euler totient function, that counts the number of fractions $a/b \in \Bbb{Q}_{\...
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0answers
23 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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1answer
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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 ...
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2answers
68 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)=...
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4answers
82 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
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0answers
27 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}\...
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1answer
55 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
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0answers
50 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 $\limsup A(n)/n$ ?
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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)|\...
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0answers
56 views

On arithmetic functions whose Dirichlet series has a special kind of abscissa of absolute convergence

For a function $f: \mathbb N \to \mathbb C$ , let $\sigma_c(f) , \sigma_a(f)$ denote the abscissa of convergence and the abscissa of absolute convergence respectively of the Dirichlet series $\sum_{n=...
1
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1answer
99 views

Modular Forms: With special congruences!

Remember the Fourier expansion of the full modular form: $$ \Delta(z) = \sum_{n=1} \tau(n)q^n $$ where $\tau(n)$ are the coefficients (Ramanujan Tau function). Ramanujan found that the coefficients ...
2
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2answers
270 views

“I am sure there are infinitely many perfect numbers”

The question Are there infinitely many perfect numbers? is a classic old unsolved problem. However, we keep finding perfect numbers (via Mersenne primes) and produce a lot of knowledge on perfect ...
2
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1answer
92 views

Show that $\sigma(n)$ is equally often even and odd

Let $\sigma(n)$ be the divisor sum of $n$: $$ \sigma(n) = \sum_{d|n} d. $$ I was interested in the parity of $\sigma(n)$ and tried to check whether $\sigma(n)$ is unexpected often even for odd $n$ ...
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3answers
67 views

Mysterious multiplicative function?

Consider $$f(n)=\prod_{p^k||n} p^{2k}(1+p^{-2})$$ Can this function be expressed by usual ones, as convolutions or directly? I do not know very well if convolution can be seen on decomposition in ...
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3answers
170 views

Long list of values of Ramanujan Tau function

In order to study Ramanujan tau function I need a long list of values. After some search in the internet I did not find a source that has a comprehensive list, i.e. more then 250 entries. I would be ...
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0answers
51 views

Partial summation $\sum_{p\leq X}a_p$ via $\sum_{n\leq X}a_n\Lambda(n)$.

Let $\Lambda$ denote the Von-Mangoldt function and $\{a_n\}_{n\in \mathbb{N}}$ a a sequence of real numbers. Assume that: $$\sum_{n\le X}a_n\Lambda(n)=O(X^\alpha)\quad\text{for some}\quad \alpha\in\...