Questions tagged [arithmetic-functions]

For questions on arithmetic functions, i.e. real or complex valued functions defined on the set of natural numbers.

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Follow-up to MSE question 3738458

This is a follow-up inquiry to this MSE question. Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
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What are the fixed points of the arithmetic derivative over the non-negative integers?

I just watched When the derivative of a number is not zero -- The arithmetic derivative by Michael Penn. I like to explore things visually and computationally, so I found this recursive implementation ...
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Full derivation inside of twin prime statement in terms of multiplicative arithmetic functions. How can the last formula be rearranged?

Let $(\cdot\mid\cdot) : \Bbb{N}\times\Bbb{N} \to \Bbb{Z}_2$ be the divisibility function which takes on the value $(x|y) = 1$ whenever $x$ divides $y$ and the value $(x|y) = 0$ whenever it does not ...
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Characterization of primes of the form $n^n+1$ by using number-theoretic functions

It is known that there is a unsolved problem related to primes of the form $n^n+1$ as is expained in page 160 of [1] (see also page 156, and the OEIS page related to this integer sequence A121270). In ...
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What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?

The following query is an offshoot of this answer to a closely related post. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect ...
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Using Mobius Inversion Formula

We are given the following: $f(n)=\prod_{d|n}g(d)$ and asked to show: $g(n)=\prod_{d|n} f(d)^{\mu(\frac{n}{d})}$ The hint given says to use logarithms Here's what I tried doing: $log(f(n))=\prod_{d|n}...
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Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the GCDs: $$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$ $$H = \...
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2 votes
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Different ways to average an arithmetic function

Consider the following ways to average a multiplicative arithmetic function $f$ over $\mathbb{N}$: Arithmetic: $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n)$ Factorized: $\displaystyle\...
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The number of solutions $(x,y)$ of the congruence $n \equiv X^2-XY+Y^2$ (mod $p^\alpha$).

Is there general formula of the solutions of the congruence? \begin{equation} n\equiv X^2-XY+Y^2 \pmod r, \end{equation} where $n\in\Bbb Z$ and $r\in\Bbb N$. If we define an arithmetic function (two ...
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Investigating $\sum \prod_{p\mid n}(1-\frac{1}{p^2})$ as $x\to\infty$

I'm investigating the behavior of the following function as $x\to \infty$: $$f(x):=\sum_{1\le n\le x}\frac{J_2(n)}{n^2}$$ where $J_k(n)$ is the Jordan's totient function $$J_k(n):=n^k\prod_{p\mid n}(1-...
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Asymptotics of the sum of squares of Von Mangoldt function values.

This question was asked in my quiz of number theory and I was not able to make any progress in it. Question: Show that $$\sum_{ n\leq x}{\Lambda(n)}^2 = x\log x- x+o(x),$$ where $\Lambda(n)$ is the ...
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Is OEIS A046346 sequence a subset $S\subset\mathbb{N}$ s.t. $\sum_S\frac{1}{n-\pi(n)}=1$?

OEIS A046346 sequence lists composite numbers divisible by the sum of their prime factors, counted with multiplicity. $$S=\Big\{n\in\mathbb{N}\space not\space prime,\space n=\prod_k p_k^{\alpha_k}\...
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Dirichlet Series of Square Full Integers.

As in the title, I want to find the Dirichlet series $F$ of the indicator function for cube full integers $f(n)=1 \iff p^3|n, \forall p|n$ and $f(n)=0$ otherwise. Since $f$ is clearly multiplicative, $...
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Counting the number of $x \in [a,b] \subset \Bbb{Z}$ such that $x^2 = k^2 \pmod d, \ \ d\mid n\#$?

Define $$ \phi(a,b,d,k)= \sum_{c\mid d \\ (c,2k)=1} \left( \lfloor\dfrac{b - x}{d}\rfloor + \lfloor\dfrac{x - a}{d}\rfloor\right).\tag{1} $$ We know that $\phi$ measures $\#\{x \in [a,b]: x^2 = k^2 \...
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Why is this alternating summation involving reciprocals of divisors always positive? A conjecture.

Conjecture. For all $n,m \in \Bbb{N}$, $$ f(n, m) := \sum_{c\mid d\mid n\# \\ \gcd(c, 2m) = 1}\dfrac{(-1)^{\omega(d)}}{d} $$ is greater than $0$. Example verification code: ...
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The formula that counts the number of averages of $2k$-separated prime pairs in the interval $[n + 2k, (n+1)^2 - 2k]$ has the following form.

Let $k \geq 1$ and $n \geq k+1$. Then the formula: $$ f(k,n) := \sum_{d \mid n\#} (-1)^{\omega(d)}\sum_{c \mid d \\ \gcd(c, 2k) = 1} \left(\lfloor \dfrac{(n + 1)^2 - 2k - x_{c,d}}{d} \rfloor + \...
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Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?

The following query is an offshoot of this post 1 and this post 2. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect numbers likely ...
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What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the divisor sum $\sigma$ is a multiplicative function.) A number $P$ is said to be perfect if $\...
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333 views

On the density of a particular subset of integers

Given a positive integer $n$ in the standard form $$n=\prod_k p_k^{\alpha_k}$$ and the arithmetic function $$f(n)=\sum_k \alpha_k p_k$$ let's define the subset $F$ of positive integers $$F=\Big\{n\in ...
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(Applied to OPN) Let $c$ and $d$ be integers, not both equal to zero. If $q$ and $r$ are integers such that $c = dq + r$, then $\gcd(c,d)=\gcd(d,r)$.

I was reading the following web resource, and found the following Lemma 8.1, which I think would be immensely useful for a computational task on odd perfect numbers (hereinafter abbreviated as OPN): ...
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If $km$ is a Descartes number with quasi-Euler prime $m$, must $\sqrt{k}$ be a squarefree palindrome?

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 numbers or positive integers. Recall that a Descartes number is an odd ...
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3 votes
2 answers
106 views

The map $m \to m\sigma(m)$ is not injective.

Let $$\tau(n) = \sum_{d \mid n}{1}$$ be the divisor function, $$\omega(n) = \sum_{p \mid n}{1}$$ be the prime divisor function, $$\varphi(n) = \#\{1 \leqslant k \leqslant n : \gcd(k,n) = 1\}$$ be ...
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1 answer
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Dirichlet series for $\zeta^3(s)/\zeta(2s)$.

I am currently studying number theory and our instructor refers to Apostol's book on Analytic number theory for the chapter Dirichlet series.In that book,there is an exercise which is as follows: Let $...
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4 answers
86 views

Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?

This question is an offshoot of this MSE answer. Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. (Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.) If $\sigma(M) = 2M$...
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3 votes
1 answer
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Show that $\sum\limits_{n\leq x} \varphi(n)=\frac{1}{2}\sum\limits_{n\leq x}\mu(n)[\frac{x}{n}]^2+\frac{1}{2}$.

I am a graduate student of Mathematics.I am now studying analytic number theory from Apostol's book.In the exercise $3$ there is a question which is as follows: If $x\geq1$,Show that $\sum\limits_{n\...
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On the reciprocal of $I(n^2) - \frac{2(q - 1)}{q}$, if $q^k n^2$ is a(n) (odd) perfect number with special prime $q$

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. A number $N$ is said to be perfect if $\sigma(N)=...
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3 answers
107 views

Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the abundancy index of the positive integer $x$ by $I(x)=\sigma(x)/...
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1 vote
1 answer
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Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II

This post is an offshoot of this MSE question. Motivation Let $x, y$ and $z$ be positive integers. Denote the sum of divisors of $x$ by $\sigma(x)$. Also, denote the deficiency of $y$ by $D(y)=2y-\...
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If $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1) n^2$, does it follow that $q n^2$ is perfect?

Let $q$ be an (odd) prime, and let $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. A number $N$ is said to be perfect if $\sigma(N)=2N$. ...
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1 vote
1 answer
121 views

On the quantity $I(q^k) + I(n^2)$ considered as functions of $q$ and $k$, when $q^k n^2$ is an odd perfect number with special prime $q$ - Part II

Hereinafter, let $N = q^k n^2$ be an odd perfect number with special/Euler prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the abundancy index of the positive integer $x$ ...
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1 vote
2 answers
72 views

Show that $\sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}$.

I am a graduate student of Mathematics. I have started reading number theory. I encountered a problem of analytic number theory. Show that $\sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{...
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2 votes
1 answer
92 views

Is there a combinatorial proof that Euler's totient function divides Jordan's totient function?

Jordan's totient function $J_{k}(n)$ is a generalization of Euler's totient function that counts the number of $k$-tuples $(a_1, \ldots, a_k)$ for which $1 \leq a_1, \ldots, a_n \leq n$ and $gcd(a_1, \...
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3 votes
2 answers
135 views

Help with "A Simpler Dense Proof regarding the Abundancy Index."

I'm reading Richard Ryan's article "A Simpler Dense Proof regarding the Abundancy Index" and got stuck in his proof for Theorem 2. The Theorem is stated as follows: Suppose we have a ...
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1 answer
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If $k=1$, then the index of an odd perfect number $p^k n^2$ (as defined by Chen and Chen (2014)) is not squarefree.

In what follows, we let $\sigma=\sigma_1$ denote the classical sum-of-divisors function. Denote the deficiency function of the positive integer $x$ by $D(x)=2(x)-\sigma(x)$, and the aliquot sum of $x$...
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2 answers
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Let $f:\Bbb{N}\to\Bbb{C}$ denotes the indicator function of squares. Express it in terms of Mobious function $\mu$.

Here $f(n)=\begin{cases} 1\ \text{if } n=m^2\text{ for some }m\in\Bbb{N}\\ 0\ \text{if otherwise} \end{cases}$ This is a multiplicative function. At first I define $g:\Bbb{N}\to\Bbb{C}$ be $g(n)$ to ...
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Is it true that $l_1(q,n) \geq g(k)$, if $q^k n^2$ is an odd perfect number with special prime $q$?

(Note: This post is an offshoot of these earlier questions: (post 1) and (post 2).) Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(...
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1 vote
1 answer
108 views

On the inequality $I(q^k)+I(n^2) \leq \frac{3q^{2k} + 2q^k + 1}{q^k (q^k + 1)}$ where $q^k n^2$ is an odd perfect number

(Note: This post is an offshoot of this earlier MSE question.) Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the ...
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Sums with von-Mangoldt function $\Lambda$

Let $\Lambda$ be the von-Mangoldt function. What is the estimate for the sum is $$\sum_{\substack{1\leq x\leq n\\1\leq y\leq n}}\Lambda(x)\Lambda(y)=\psi(n)\psi(n)\sim n^2.$$ $\psi$ is the Chebyshev ...
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4 votes
0 answers
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Two numbers with a given difference having the same number of divisors

So, it is required to prove that for each natural $k$ there are two natural numbers with a difference $k$ having the same number of divisors. For example, for the case $k=27$, the pair $(18,45)$ is ...
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A problem on sum of positive divisors

I am learning a paper about deficient-perfect numbers. I found a equation that I still can't understand it. We start from definition: For a positive integer n, let σ(n) denote the sum of the positive ...
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$ (1*...*1)(n) =\# \{(m_1,...,m_k)\in (\mathbb{N}\setminus \{0\})^k : m_1...m_k = n\} $

We call an arithmetic function any element of $\mathcal{F} (\mathbb{N}\setminus \{0\} , \mathbb{C})$. We endow $\mathcal{F} (\mathbb{N} \setminus \{0\}, \mathbb{C})$ binary operation convolution and ...
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2 votes
2 answers
57 views

Set of Integers with a Given Abundancy Index

Let $\sigma (n)$ be the sum of divisors of $n$. Define the abundancy index of $n$ to be $I(n)=\frac {\sigma(n)}{n}=\frac ab$ with $a,b$ coprime integers. For a given limit $L$ and coprime integers $a,...
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If $q^k n^2$ is an odd perfect number with special prime $q$, is $\gcd(\sigma(q^k),\sigma(n^2))=1$ equivalent to $k=1$?

Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Here is my: QUESTION If $q^k n^2$ is an odd perfect number with special prime ...
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1 answer
55 views

Dirichlet inverse for $\left\{1,0,-1,0,1,0,-1,0,1,0,-1,0,\ldots\right\}$

I am looking for the Dirichlet inverse of $\left\{1,0,-1,0,1,0,-1,0,1,0,-1,0,\ldots\right\}$ or equivalently $$ f(n)=\frac{ {i^{n-1}+(-i)^{n-1}}}{2}. $$ It is an interesting inverse, it seems always ...
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  • 502
0 votes
1 answer
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Asymptotic notation and big-O notation.

We had a course on elementary number theory in a postgraduate course. Our instructor started the course with arithmetic functions. He introduced Euler's summation formula which states that if $f$ is ...
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1 answer
70 views

How to prove this upper bound of generalized Mangoldt function?

Define $\Lambda_k(n,x)=\sum\limits_{d|n}\mu(d)\left(\ln\frac xd\right)^k$, where $n$ is a positive integer and $x$ is a positive real number. Let $n=p_1^{a_1}...p_r^{a_r}$, prove that $\Lambda_k(n,x)\...
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15 votes
1 answer
438 views

Polynomials whose fractional part behaves like a logarithm

For a given integer $m$, I'm looking for a classification of all polynomials $P$ with rational coefficients satisfying the logarithm-like condition $$P(ab)=P(a)+P(b) \pmod 1$$ for any integers $a, b$ ...
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1 vote
2 answers
127 views

Prove $\sum_{d | n} \mu(d) (\log(d))^2=0$ [duplicate]

If $n$ is a positive integer with more than 2 distinct prime factors, how to prove that $\sum_{d | n} \mu(d) (\log(d))^2=0$? I struggle on how to continue from this. Suppose $n=p_1 p_2 ... p_r$, ...
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2 votes
1 answer
168 views

On odd perfect numbers $q^k n^2$ and the deficient-perfect divisor $q^{\frac{k-1}{2}} n^2$ - Part III

(Preamble: This question is an offshoot of this answer by mathlove to an earlier post.) Let $\sigma(x)=\sigma_1(x)$ denote the classical sum of the divisors of the positive integer $x$. If $\sigma(m)=...
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2 votes
2 answers
77 views

On improving the upper bound $I(m^2) \leq \frac{2p}{p+1}$, if $p^k m^2$ is an odd perfect number with special prime $p$

(Preamble: This question is an offshoot of this earlier MSE post.) Let $p^k m^2$ be an odd perfect number (OPN) with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$. ...
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