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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if such counter example to Lehmer's totient problem exists then could we have more counter examples?

Lehmer's totient problem asks whether there is any composite number $n$ such that Euler's totient function $φ(n)$ divides $n − 1$. which it is unsolved problem or we may reformulate that question as : ...
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34 views

An identity of Arithmetic Functions

Problem: Show that for all positive integers $n$, $$ \sum_{a=1, (a,n)=1}^{n} (a-1, n) = d(n)\phi(n)$$ where $(a, b)$ stands for $\text{gcd}(a, b)$ and $d, \phi$ are the divisor and Euler's totient ...
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28 views

Summation of certain divisible numbers

Let $C(N,m)$ be the number of positive integers $\le N$ which are relatively prime to $m$. It can be found by following equation \begin{align} C(N,m)=\sum_{d|m}\mu (d)\left\lfloor\frac{N}{d}\right\...
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On odd perfect numbers and a GCD - Part III

(Note: This post is an offshoot of this earlier MSE question.) In what follows, we let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. We also let $D(x)=2x-\sigma(x)$ denote the ...
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Why does the divisor-counting function appear in bounds for Kloosterman sums?

Given integers $m,n$ and $c \geq 2$, the Kloosterman sum is defined as $S(m,n;c) = \sum_{k \in (\mathbb{Z}/c\mathbb{Z})^{\times}}{e^{\frac{2i\pi}{c}(mk+nk^{-1})}}$, where $k^{-1}$ is the reciprocal of ...
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29 views

About the characterization of solutions of an equation that involves particular values of the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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1answer
20 views

A total function is representable iff it is weakly representable

The book A Friendly Introduction to Mathematical Logic - 2nd Edition by Christopher C. Leary and Lars Kristiansen gives the following proposition without proof: Proposition 5.3.6. Suppose that $f$ ...
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1answer
51 views

What is meant by $\sum_{d \le x}f(d)$ in this equation?

Wikipedia's page (here) on the average order of arithmetic functions gives the following as a means of finding such an order using Dirichlet Series: Define $f$ as an arithmetic function on $n$, and ...
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Proving identity using Dirichlet L functions

I'm trying to prove the following identity using Dirichlet L functions : ${\displaystyle \sum _{d\mid n}\varphi (d)=n}$ I have shown proved that the Dirichlet Series of $\varphi (n)$ equals to ${\...
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104 views

Why is this arbitrary-looking identity of arithmetic functions “obvious”?

This question is about exercise 2.31 in Apostol's Introduction to Analytic Number Theory. The question asks us the following: if $f$ is a multiplicative arithmetic function, and $g$ is a completely ...
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52 views

Finding $f$ with Möbius inversion formula

How to find an arithmetic function $f$, when the summatory function $F$ of $f$ is given by $F(n)=\begin{cases} 1 & \mathrm{if}\ n \mathrm{\ is \ a \ square \ number} \\ 0 & \mathrm{...
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Showing that $\sum_{m=1}^{n}{\sigma(m)}=\sum_{k=1}^{n}{k\cdot \left\lfloor \frac n k\right\rfloor}$

Let $n \in \mathbb{N}$. $\sigma$ is an arithmetic function and $\sigma(n)$ is the sum of the (positive) divisors of $n$. How to show that $\sum_{m=1}^{n}{\sigma(m)}=\sum_{k=1}^{n}{k\cdot \left\...
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78 views

Can this inequality involving the deficiency and sum of aliquot divisors be improved?

In what follows, we let $n > 1$ be a positive integer. The classical sum of divisors of $n$ is given by $\sigma_1(n)$. Denote the abundancy index of $n$ by $I(n)=\sigma_1(n)/n$. Denote the ...
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29 views

Proof that for all $m,n,l \in \mathbb{N}_0$, $m\leq n \iff m+l \leq n+l$.

I am trying to prove the claim: "for all $m,n,l \in \mathbb{N}_0$, $m\leq n \iff m+l \leq n+l$". In what follows, $s_m$ for (some natural $m$) is the unique function that exists by the recursion ...
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48 views

If an arithmetic function $f$ satisfies $f(mn) \leq f(m)f(n)$ (whenever $\gcd(m,n)=1$), is $f$ weakly multiplicative or submultiplicative?

From the preprint On sums of the small divisors of a natural number (Lemma 1, page 2) by Douglas E. Iannucci: We observe here that the function $a(n)$ is not multiplicative. It is, however, ...
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Functions that link to divisor function

I am interested in the following problem: Do there exist a function $f:\mathbb{N}\to\mathbb{N}$ such that $$f(f(n))=\sigma_0(n)$$ My progress was the following (for sake of simplicity, we denote $\...
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106 views

Dirichlet Series in analytic number theory

I have a question about Abscissa of Convergence of Dirichlet series. The question is ; "Let $\sigma_{1}$ and $\sigma_{2}$ be real numbers with $\sigma_{1} \leq \sigma_{2} \leq \sigma_{1}+1 .$ ...
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1answer
142 views

A question about Dirichlet series [closed]

i have the following question given 2 options as i) and ii) Let $f(n)$ be the unique positive real-valued arithmetic function that satisfies $\sum_{d | n} f(d) f(n / d)=1$ for all $n$ . (i.e., $f$ ...
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1answer
26 views

Show this congruence holds

I need to show that if $d|p-1$, $d<p-1$ and $g$ is a primitive root modulo $p$ then, $$\sum_{l=1}^{(p-1)/d} g^{dl}\equiv 0\pmod{p}.$$ I have a gut feeling it has something to do with the fact that $...
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80 views

When does $\gcd(m,\sigma(m^2))$ equal $\gcd(m^2,\sigma(m^2))$? What are the exceptions?

(This question is related to this earlier one.) Let $\sigma(x)$ be the sum of divisors of the positive integer $x$. The greatest common divisor of the integers $a$ and $b$ is denoted by $\gcd(a,b)$. ...
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On odd perfect numbers and a GCD - Part II

Note that $\gcd(\sigma(q^k),\sigma(n^2))=i(q)=\gcd(n^2,\sigma(n^2))$ if and only if $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$. I have therefore undeleted this question, in view of this closely ...
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120 views

Prove that for every $ϵ>0$ there exist infinitely many natural numbers $n$, such that $τ(n)\geq 2^{(1-ϵ)(\log(n)/\log(\log(n))}$.

Prove that for every $ϵ>0$ there exist infinitely many natural numbers $n$, such that $$\tau(n)\geq 2^{(1-ϵ)\frac{\log(n)}{\log\log(n)}}$$ It is obvious to look at squarefree naturals of the form $...
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Attempt to get a characterization of even perfect numbers from an equation involving the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(n)$ for integers $n\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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1answer
51 views

Is there a number $\mathscr{D}_2 \neq \mathscr{D} = {{3003}^2}\cdot{22021}$ satisfying a certain condition?

(Note: This question is tangentially related to this earlier one.) 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 ...
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1answer
30 views

Number of integers for which $\Omega=k$

for every $x>0$, I'm trying to find out the number of positive integers $n\leq x$ such that $\Omega(n)=k$ for all $k\geq 1$ in $N$, where $\Omega(n)=\Omega(p_1^{\mu_1}\cdot\ldots\cdot p_j^{\mu_j})=\...
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1answer
39 views

An improved inequality for the deficiency function when $\gcd(x,y)=1$, $x > 1$, and $y > 1$

(The following is an attempt to improve on the result contained in this MSE question.) Let $\sigma(x)$ be the sum of the divisors of a (positive) integer $x$. (For example, $\sigma(2) = 1 + 2 = 3$.) ...
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68 views

An inequality for the sum-of-aliquot-divisors function

In what follows, we shall assume that $a$ and $b$ are relatively prime. (That is, $\gcd(a,b)=1$ holds.) It is known that the inequality $$\sigma(ab) \leq \sigma(a)\sigma(b)$$ holds for the sum-of-...
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How to prove $\sum_{d|n} \mu(\frac{n}{d}) P(d) = 1$?

Background I came up with an unorthodox proof of the following: $$\sum_{d|n} \mu(\frac{n}{d}) P(d) = 1$$ Where $n \neq 1$, $P(d)$ is a function which counts the number of primes of $d$ and $\mu$ ...
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1answer
35 views

An absolute upper bound for $\omega(n)$

Let $\omega(n)$ the number of distinct prime numbers in the factorisation of $n$. Many results are known about the average of $\omega(n)$. One of them is that $\omega(n)\sim \log \log n$ for almost ...
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49 views

Lower bound related to the number of distinct prime numbers

Let $\omega(n)$ be the number of distinct prime factors of $n$ (without multiplicity, of course). I know some results about average of $\omega(n)$. But I didn't find any result about the following: ...
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How to estimate the mean value of $4^{\Omega(n)}$?

How to get an asymptotic formula of $$ \sum_{n\leqslant x} 4^{\Omega(n)},$$ where $\Omega$ is the counting functions of the total number of prime factors of $n$, taken with multiplicity. I find a ...
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1answer
80 views

Prove that $J_r(n) = n^r\prod_{p|n}\left(1− \frac{1}{p^r}\right)$.

For $r ≥ 1$, let $J_r(n)$ be Jordan's totient function, i.e., the number of $r$-tuples of integers $a_j$ with $1 ≤ a_j ≤ n$ for $j = 1,\ldots,r$, and $(a_1,\ldots,a_r,n) = 1$. Prove that $$J_r(n) = n^...
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$(-1)^n$ times Mobius function

Let $f : \mathbb{N} \rightarrow \mathbb C$ be defined by $f(n) = (-1)^n$ and let $g : \mathbb{N} \rightarrow \mathbb C$ be the Mobius function. Define $h : \mathbb{N} \rightarrow \mathbb C$ be $h(n) =...
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24 views

How to reverse extrapolate values $Q$ and $C$ from arithmetic function?

I'm attempting to create a function that causes $Q$ to increase by $1$ whenever $C$ increases by $Q$. The progression of $C$ can be mapped by using $(n/2)*(2Q+(n-1)d)$, where $d = 1$ and $n= Q-9$. ...
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1answer
41 views

Is this function an injection of $\mathbb{N} \to \mathbb{R}$?

This comes from an earlier question of mine. It seems unapproachable. Is the arithmetic function $$f(n) := \sum_{k=1}^n \frac 1{\sin(k)}$$ An injection? I think the answer is yes.
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Get rid of duplicated values with basic math

I've been struggling for some time with this. Is there any way to sum only unique values if you can only use data in table below and: Sum Average number of rows number of unique rows Column 'total ...
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108 views

Why tetration and more are not common in nature?

I didn't see this question in here but it was asked in quora and it was interesting to me that no one had any satisfying answers. some people suggested that it's because exponentiation describes the ...
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109 views

Sum with the von-Mangoldt function: $\sum_{1\leq x\leq n}\Lambda(x)^4$

Let $Λ$ be the von-Mangoldt function. then What is the estimate for the sum $\sum_{1\leq x\leq n}\Lambda(x)^{4}$? Is this $\sum_{1\leq x\leq n}\Lambda(x)^{4}\sim n\log^3n$ also what can we say about ...
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Name of Möbius type operator?

Does anyone know if the Möbius / Dirichlet - type operator: $$f \mapsto \sum_{d|n} f(d)$$ that (among other things) appear in the Möbius inversion formula has a name? Would it be fair to call it the ...
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1answer
51 views

Implications of $q \neq 1049$ when $q^k n^2$ is an odd perfect number

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 $\gcd(q,n)=1$. In what follows, we let $$I(x) = \frac{\sigma(x)}{x}$$ denote the ...
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49 views

On the biconditional $I(n^2) = 2 - \frac{5}{3q} \iff (k = 1 \land q = 5)$, where $q^k n^2$ is an odd perfect number

MOTIVATION Let $N$ be an odd perfect number given in the so-called Eulerian form $$N = q^k n^2,$$ i.e., $q$ is the special / Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. ...
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1answer
94 views

Showing that the limit is $1/\varphi(q)$

Fix $a \in \mathbb{Z}$, $q \in \mathbb{N}$ and define an arithmetic function $f$ as follows: $f(n)= \log n$ if $n$ is prime, $0$ otherwise. Can we show that the following limit $\displaystyle \lim_{N\...
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55 views

Do these arithmetic functions have an infinite number of integer zeros?

This question assumes the following definitions. (1) $\quad a(n)=\sum\limits_{d|n}\mu(d)\,\mu\left(\frac{n}{d}\right),\qquad \frac{1}{\zeta(s)^2}=\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}\qquad$(see ...
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1answer
63 views

determining a convolution of an arithmetic function

Let be $ \lambda: \mathbb{N} \rightarrow \mathbb{C}$ be an arithmetic function $$ \lambda (n) := (-1)^{e_1+\dots+e_r} $$ where $p_1^{e_1}...p_r^{e_r} $ is the prime factorization of $n$ and it is $ \...
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1answer
39 views

A possible disproof for the Descartes-Frenicle-Sorli Conjecture on odd perfect numbers

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 $D(z):=2z-\sigma(z)$, and the sum of the aliquot divisors by $s(z):=...
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1answer
27 views

arithmetic function invertible concerning convolution?

Let be $$ \phi: \mathbb{N} \rightarrow \mathbb{C} $$ an arithmetic function. How can I show that $ \phi $ is only invertible concerning convolution if $ \phi (1) \neq 0 $ holds? So ..: $\phi $ ...
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2answers
52 views

Does normal order imply average order?

Let $g$ be a normal order of an arithmetic function $f$, i.e. for every $\epsilon > 0$, $$\left| \frac{f(n) - g(n)}{g(n)} \right| < \epsilon$$ holds for almost all $n \in \mathbb{N}$. Does this ...
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2answers
62 views

Proving that $g(1) = 1$, where $g$ is a multiplicative arithmetic function

I'm having some trouble understanding a simple problem about an arithmetic function. The problem is simply to answer true or false that $g(1) = 1$, assuming $g$ is multiplicative and $g(n)$ $\neq 0$...
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1answer
53 views

Prove that $\sum_{n\leq x}{d(\phi(n))} > x \log{x}$

Prove that $$\sum_{n\leq x}{d(\phi(n))} > x \log{x},$$ where $d(n)$ is the number of factors of $n,\phi(n)$ is the Euler's totient function. From Wikipedia, I know that $$\phi(n)>\frac{n}{e^\...
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1answer
58 views

Natural numbers $n$ satisfying $\mu(n+1)+\mu(n+2)+\cdots+\mu(n+2019)=2019$

Determine all nautral numbers $n$ satisfying $$\mu(n+1)+\mu(n+2)+\cdots+\mu(n+2019)=2019$$ Where $\mu (n) = (-1)^{\omega(n)}$ if $n$ is square free, and $\mu(n)=0$ otherwise. $w(n)$ denotes ...

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