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 question 3121498, asked in February 2019

Let $n = p^k m^2$ be an odd perfect number given in Eulerian form (i.e. $p$ is the special/Euler prime satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$). Denote the classical sum of ...
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28 views

How to determine the number of integer pairs $n_1,n_2$ such that $n_1+2n_2=n$, with $n,n_1,n_2=0,1,2,3,…$?

In the context of a Quantum Physics problem with degenerate energy levels, I need to find the number of different pairs of integers $(n_1,n_2)$ such that $n_1+2n_2$ gives the same value $n$, with $n,...
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1answer
28 views

Is there any integer $x$ such that $2^n$ divides $3^n(x+1)$ for all integers $n$? [closed]

I am wondering whether: $$\exists x \in \mathbb{N}^* / \, \forall n \in \mathbb{N}^*,\, 2^n\mid 3^n(x+1)$$ I re-wrote it as $$2^n \leq 3^n(x+1)$$ but it doesn't seem like a good approach. Any ideas?
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2answers
56 views

Suppose $f$ is continuous on $[-1;1]$ such that $x^2 +f^2(x) = 1$ for all $x$. Show that $f(x) = \sqrt{1-x^2}$ or $f(x) = -\sqrt{1-x^2}$ for all $x$

Suppose $f$ is continuous on $[-1;1]$ such that $x^2 +f^2(x) = 1$ for all $x$. Show that $f(x) = \sqrt{1-x^2}$ or $f(x) = -\sqrt{1-x^2}$ for all $x$ This is my idea: $x^2 + f^2(x) = 1 \Leftrightarrow ...
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0answers
34 views

Zero divisors in the Dirichlet ring

I'm trying to determine if the ring $(\mathbb{A}, +, *)$ is an integral domain, where $\mathbb{A}$ is the set of arithmetic functions and $*$ is the Dirichlet convolution. To do so, I'm trying to ...
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32 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 $q$

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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3answers
86 views

Applying a criterion on deficient numbers to the proper factors of an odd perfect number

Hereinafter, let $I(x)=\sigma(x)/x$ denote the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$. Denote the deficiency of $x$ by $$D(...
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1answer
49 views

A proof (?) for $k = 1 \implies q \neq 5$, 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 given in Eulerian form (i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$). Inspired by mathlove's answer to ...
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1answer
121 views

On bounds for the quantity $n^2 / D(n^2)$ when $q^k n^2$ is an odd perfect number with special prime $q$ and $k > 1$

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form (i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$). In this post, I would like to ...
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50 views

Identifying an arithmetic function

Define a function $f: \mathbb N \to \mathbb N$ by $f(p^k) = p^k-1$ on prime powers and extended to full natural numbers by defining $f$ to be a multiplicative function. Then $f$ agrees with the Euler-...
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45 views

On improving $D(q^k)/q^k > (q-2)/(q-1)$, if $D(x)=2x-\sigma(x)$ and $q$ is a prime number

It is known that, for $q$ prime and $k$ a positive integer, $$I(q^k) = \frac{\sigma(q^k)}{q^k} = \frac{1 + q + \ldots + q^k}{q^k} = \frac{q^{k+1} - 1}{q^k (q - 1)},$$ where $\sigma(x)=\sigma_1(x)$ is ...
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1answer
57 views

Is there an odd $x$ such that $2x^2 \equiv 0 \pmod {\sigma(x)}$ and $\sigma(x^2) \equiv 0 \pmod {\sigma(x) - 1}$?

CONTEXT This question is a result of considerations stemming from this closely related MO question. INITIAL QUESTION My question is as is in the title: Is there an odd $x$ such that $$2x^2 \equiv 0 \...
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1answer
26 views

Sum of an arithmetic function over the divisors of an integer is multiplicative

I am trying to prove the following claim: Let $f$ be an arithmetic function and, for $n \in \mathbb{Z}$ with $n > 0$, let $$ F(n) = \sum_{d\ \vert\ n,\ d > 0}f(d) $$ If $F$ is multiplicative, ...
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1answer
113 views

On a conjectured upper bound for $k=\nu_q(N)$, if $N=q^k n^2$ 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 $x$ as $I(x)=\sigma(x)/x$ ...
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0answers
16 views

Explanation of something related to solving $\phi(x) =m$ [duplicate]

Let $\phi(x)=m$, where $\phi(x)$ is Euler’s phi function, And $x=\prod_{i=1}^{n}p_i^{a_i}$ Thus: $$\phi(x)=\phi\left(\prod_{i=1}^{n}p_i^{a_i}\right)= \prod_{i=1}^{n}p_i^{a_i}-p_i^{a_i-1} =m$$ $$\iff m=...
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35 views

Sum of reciprocals of the sum of divisors

For a few days, I've been unsuccessfully trying to solve this problem: Let $\sigma(n)$ be the sum of divisors of the integer $n$ (For example: $\sigma(10) = 10 + 5 + 2 + 1$). Find an asymptotical ...
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41 views

A problem with regards to the sum of the digits of a number

The problem says the following: Let $S(n)$ be the sum of digits of some number $n \in \Bbb N^*$. Find a natural number $k$, such that $S(k)=2017 \times S(3k) $. (The sum of the digits of a number is ...
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1answer
34 views

Explanation of the proof of $\phi(n)\cdot \frac{n}{2}=\sum_{a\in R}a$

In my number theory textbook, there is a theorem about Euler's Phi Function it says: A Theorem: If $R$ is a reduced residue system modulo $n$: $R=\{a\in C \mid \gcd(a,n)=1\}$ Where $C$ is a complete ...
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2answers
57 views

The formula to calculate Euler Phi function

I know that $$\phi(m)=m\prod_{i=1}^{n}\left(1-\frac{1}{p_i}\right)\text{ Where }m=\prod_{i=1}^n p_i^{a_i}$$ But when i tried to find a formula of $\phi(n)$ i got this: $$\phi(m)=\phi \left(\prod_{i=1}...
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3answers
64 views

Prove that $\sigma(n)/n = \sum_{d\mid n} 1/d$ [duplicate]

Prove that $$\frac{\sigma(n)}{n}=\sum_{d\mid n}\frac{1}{d} ,\forall n\in \mathbb N^*$$ It’s from a number theory textbook , i’ve tried to use to formula of $\sigma(n)$, which is : $$\prod_{i=1}^{n} \...
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1answer
62 views

Example of a graded ring with respect to any completely additive arithmetic function - is it isomorphic to anything?

Let $\varphi : \Bbb{N} \to \Bbb{N}$ be a completely additive arithmetic function, i.e. $\varphi(nm) = \varphi(n) + \varphi(m)$ for all $n, m \in \Bbb{N}$. For example, $\Omega(n) = $ the total number ...
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0answers
46 views

Generalised Möbius inversion formula.

Lets recall the Möbius inversion formula: If we have two arithmetic functions $f,F$, such: $$F(n)=\sum_{d|n}f(d)$$ then we have: $$f(n)=\sum_{d|n}F(d)\mu(\frac{n}{d})$$ Suppose now that we have ...
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497 views

On $\frac{D(n^2)}{s(q)} \geq \frac{2n^2}{\sigma(q)} \geq \frac{\sigma(n^2)}{q} \geq \frac{2s(n^2)}{D(q)}$ where $q^k n^2$ is an odd perfect number

Let $N=q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Descartes, Frenicle, and subsequently Sorli conjectured that $k=1$ always ...
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1answer
75 views

On the inequality $\frac{\sigma(n^2)}{q^k} < \frac{n^2 - q^k}{C}$ where $C>1$ and $q^k n^2$ is an odd perfect number - Part II

(Note: This is a continuation of this earlier 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 $\gcd(q,n)=1$. (In particular, ...
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2answers
358 views

Does the following lower bound improve on $I(q^k) + I(n^2) > 3 - \frac{q-2}{q(q-1)}$, where $q^k n^2$ is an odd perfect number?

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 abundancy index $$I(x)=\frac{\sigma(x)}{x}$$ where $\sigma(x)$ ...
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1answer
30 views

If two multiplicative functions are close, are their Dirichlet inverses necessarily close?

Let $f$ be a multiplicative arithmetic function such that $\sum_{n = 1}^{\infty} |f(n)|$ converges. Given some $\epsilon > 0$, can we necessarily choose some $\delta$ such that for all arithmetic ...
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1answer
246 views

Revisiting questions 3888565 and 3894925

The topic of odd perfect numbers likely needs no introduction. I would like to revisit these two questions: Is it possible to improve on the bound $D(q^k) < \varphi(q^k)$ if $k>1$? On the ...
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1answer
65 views

A sum involving the Mobius function and a product over prime factors

This is a follow-up question to: What is known about partial sums involving squares of the totient function? Using the hint given by Daniel Fischer, I was able to express the sum posed in the question ...
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3answers
101 views

Explanation of the statement of the fundamental theorem of arithmetic.

It's from the book Mathematics Made Difficult. If $L^+(P,N_0)$ is the set of functions $f:P\rightarrow N_0$ with a property such that $$\exists\; n_0 \in N_0 \; \forall \; p \in P \;$$ $$ p\ge n_0 \...
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0answers
69 views

What is known about partial sums involving squares of the totient function?

The behaviour of the sum: $$\displaystyle\sum_{n \leq x} \frac{\varphi(n)}{n}$$ is well-studied. However, I am interested in the behaviour of the following sum, and haven't been able to find any ...
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1answer
42 views

Check if an arithmetic function is (completely) multiplicative

In Paul J. McCarthy's book about arithmetic functions, he constructs a completely multiplicative function $g$ by setting $g(p)$ equal to a root of the equation $$X^2+f^{-1}(p)X+f^{-1}(p^2)=0, $$ where ...
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1answer
21 views

Inclusion-exclusion of values of $\Omega$ (the arithmetic function).

For any number $n \in \Bbb{Z}$, define $$ f(n) = \Omega(n) - \sum_{q \mid n} \Omega(n/q) + \sum_{q,q' \mid n}\Omega(n/(qq')) - \dots \pm \Omega(1),$$ where each sum is taken over only prime divisors ...
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3answers
96 views

Showing that an arithmetic function is multiplicative

Let $f$ be an arithmetic function that counts the number of consecutive integers between $1$ and $n$ (inclusive) such that both integers are coprime to $n$. More formally, $$ f(n) = \sum_{\substack{1 \...
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1answer
34 views

Asymptotic behavior of $\sum\limits_{n \leq x}\frac{\phi(n)}{n}$

I'm working with the asymptotic behavior of square-free integers and have already proven that if $f$ is an arithmetic function with $F(n) := \sum\limits_{d \mid n}f(d)$, then \begin{align*} \sum\...
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2answers
78 views

Sum of $\frac{\Lambda(n)}{\log(n)}$ over divisors

I am wondering if $\sum_{d\mid n}\frac{\Lambda(d)}{\log(d)}$ (which evaluates to $1/k$ for $d=p^k$ and $0$ otherwise) has any interesting significances or bounds. For $n = p_1^{k_1}\dots p_s^{k_s}$, ...
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1answer
86 views

Convergence of the double series $\sum_{d|n} \mu(d) x_{n}$

Let $\sum x_n$ be a power series which behaves sufficiently nicely, for example, absolutely convergent. Can we deduce that the double series $$ \sum_{d|n} \mu(d) x_{n} $$ converges in the sense ...
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1answer
29 views

Asymptotics for convolutions of arithmetic functions

I would like to know the asymptotics for $n \to \infty$ of the two functions $$\sum_{d \mid n} \frac{\mu(d)}{d^2}n^2$$ and $$\sum_{d \mid n} \frac{\mu^2(d)}{d} n.$$ I see nothing else to do than to ...
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0answers
23 views

$\lim_{n \to \infty}\frac{\log{(\prod_{a_i\leq n} a_i)}}{(\text{number of }a_i\leq n)\log{n}}$ for $\{a_n\}$ a subsequence of natural numbers

This is a generalization of Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$. . Precisely, let $\{a_n\}$ be a subsequence of $1, 2, 3, ...$. Let $A(n):=\sum_{a_i\leq n}1$ counts the number of $...
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0answers
46 views

Sum of inverse of a multiplicative function

I stumbled upon the following problem, while trying to come up with a recreational math question. Let $n$ be a positive integer with factorization $n=2^a\prod_{i=1}^{k}p_i^{e_i}$ Define the arithmetic ...
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2answers
126 views

Number of ways to write $2n$ as sum of two primes is unbounded

For all $k>0$, we are to show that there exists an even number $2n > 0$ that can be written as sum of two primes in at least $k$ ways. ($8=3+5=5+3$ counts as one way) I tried to use pigeonhole ...
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2answers
66 views

Is it possible turn the Dirichlet ring into a Banach algebra?

The set of all arithmetic functions $f:\mathbb{Z}^{+}\to\mathbb{C}$, under pointwise addition and Dirichlet convolution, is a commutative ring, not all functions are Dirichlet invertible. So my ...
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1answer
51 views

Dirichlet convolution of the small prime omega function and the Mobius function

I have seen that: $$(\omega\star\mu)(n)=\sum_{d\vert n}\mu(d)\omega\left(\frac{n}{d}\right)=\begin{cases}1 & n\ \text{is prime}\\ 0 &\text{otherwise} \end{cases}$$ where $\mu(n)=\delta_{\omega(...
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2answers
73 views

Maximal order of magnitude of Prime Omega Function

Let $\omega(n)=\sum_{p|n}1$ be the prime omega function. Prove that for any $\epsilon > 0$, There exists $N>0$ such that for all $n > N$, $$\omega (n)<\frac{(1+\epsilon)\log{n}}{\log{\...
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2answers
84 views

Asymptotic formula for Mean of Sum of Power of Divisors $\frac{ \sum_{i=1}^{n}\sigma_v(i)}{n}\sim\frac{n^v\zeta(v+1)}{v+1}$

Question: Define the sum of $v$-powers of divisor $\sigma_v(n)=\sum_{d|n}d^v$ for $v \in \mathbb{R}$. Prove that for all $v>0$, $$\frac{ \sum_{i=1}^{n}\sigma_v(i)}{n}\sim\frac{n^v\zeta(v+1)}{v+1}$$ ...
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1answer
234 views

Is it possible to improve on the bound $D(q^k) < \varphi(q^k)$ if $k > 1$?

The problem is as is in the title: Is it possible to improve on the bound $$D(q^k) < \varphi(q^k)$$ if $k > 1$? Here, $q$ is a prime number and $k$ is a positive integer. The (deficiency) ...
1
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0answers
47 views

Proving a relation between sum of reciprocal of divisors and $\sigma(n)$

Prove that $\sum_{d\mid n}\dfrac{1}{d}=\dfrac{\sigma(n)}{n}\ \forall\ n\geq 1, n\in \mathbb Z$ My question and my approach is a lot similar to this question and a bit different from this question ...
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1answer
108 views

Is this expression bounded when $p \equiv 1 \pmod 4$ is prime and $t \equiv 1 \pmod 4$?

Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. The problem under consideration is as in the title: Is the following expression bounded when $p \equiv 1 \pmod 4$ is prime and $...
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1answer
223 views

The definition of Arithmetic Function

Consider following functions (written on Page 120 of the book "Summing It Up" by Avner Ash and Robert Gross, 2016): $$ a(n)= \begin{cases} 0 & \text{if $n$ is not prime and} \\ ...
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0answers
51 views

How to deduce the sign changes?

Let $f$ be an arithmetical function. Suppose that there exists an integer set $A$ compromises the integers $n$ for which $f(n)>0$ and a set $B$ of integers $n$ such that $f(n)<0$ and a set $C$ ...
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1answer
92 views

Implying form the Symmetry of the sum of von Mangoldt function

In the article "On the Selberg-Erdos Proof of The Prime Number Theorem" by Ashvin A. Swaminathan, page 5, it is written - By rearranging the claimed equality, it suffices to show that $$\...

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