Questions tagged [multiplicative-function]

This tag is for questions relating to multiplicative functions which are arise most commonly in the field of number theory.

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2
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2answers
42 views

Prove of $\prod_{d|n} (\mu(d)(\mu(d) + 3) + 4) = 4^{d(n)}$

Found an interesting relation: $$\prod_{d|n} (\mu(d)(\mu(d) + 3) + 4) = 4^{d(n)}$$ where $\mu(n)$ is a Möbius function and $d(n)$ is a divisors count. I think this should be something known. The prove ...
0
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0answers
46 views

How does this multiplication work?

I'm studying the CL signature that how can it works. However, I cannot understand how equation (42) is calculated. What is the last three factors? The value T is as follows: The full paper link is ...
2
votes
3answers
65 views

find all $n$ such that $\varphi(\sigma(2^n)) = 2^n$

Problem: Find all positive integers $n$ such that $\varphi(\sigma(2^n)) = 2^n$, where $\varphi(n)$ is Euler's totient function and $\sigma(n)$ is the sum of all divisors of $n$. I know that $\sigma(2^...
1
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2answers
50 views

Sum of divisors function inequality

Prove that if $n<m$ and $n$ divides $m$, then $\frac{\sigma(n)}{n} < \frac{\sigma (m)}{m}$, where $\sigma(x)$ denotes the sum of all the divisors of $x$. I know that $\sigma (x)$ is ...
0
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1answer
28 views

Show $\sum_{c|n} \mu(c)f(c) = \{1-f(p_1)\}\{1-f(p_2)\} \dots \{1-f(p_r)\}$ [closed]

$n=p^{k_1}_1p^{k_2}_3...p^{k_r}_r$ and f is multiplicative function.I have tried convolution but it seems not solving.
0
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0answers
20 views

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 ...
0
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1answer
53 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, ...
3
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1answer
49 views

A problem regarding the product of all the elements of $U_n$ for some selected $n$

$\mathbf {The \ Problem \ is}:$ Find the product of all elements of the multiplicative group $U_n$ where $n=p^2q$ and $p^2$ for distinct primes $p$ and $q ?$ $\mathbf {My \ approach} :$ Actually, ...
3
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1answer
88 views

Estimate for $\sum_{n\leq x}2^{\Omega(n)}$

I need some help to find a mistake in my proof. I have to prove that $\sum_{n\leq x}2^{\Omega(n)}\sim cx\log^2x$ for $x\rightarrow+\infty$, where $\Omega(p_1^{k_1}\cdot\ldots\cdot p_j^{k_j})=k_1+\...
1
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1answer
34 views

Reference request: Axiomatic treatment of multiplicative functions?

I'm currently reading Apostol's analytic number theory, Chapter 2 on multiplicative functions. While the current exposition is nice, I can't help but feel that there has to been some algebraic ...
2
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0answers
24 views

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 ...
1
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1answer
36 views

Discrete distributions such that $P(XY=ab) = P(X=a)P(Y=b)$

I invented a fun exercise: Suppose X and Y are discrete independent random variables over $\mathbb{N}$ such that $$\forall a, b \in \mathbb{N}, ~~\mathbb{P}(X*Y = a*b) = \mathbb{P}(X=a) *\mathbb{P}(...
4
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0answers
135 views

Show $\gcd(a,b)=1$ implies $\varphi(a\cdot b)=\varphi(a)\cdot\varphi(b)$

This means that if $\gcd(m, n) = 1$, then $φ(mn) = φ(m) φ(n)$. (Outline of proof: let $A, B, C$ be the sets of nonnegative integers, which are, respectively, coprime to and less than $m, n$, and $mn$; ...
0
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0answers
30 views

Proving an identity with Euler products

I need to show that if $f$ is a multiplicative function supported on the squarefree integers defined by $$f(p)=\frac{L}{\sqrt{p}\log p}$$ if $L^2\leq p \leq \exp((\log L)^2)$ and $f(p)=0$ otherwise,...
2
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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 $g(n)$ $\neq 0$...
0
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0answers
32 views

Ideal norm always super-multiplicative?

Let $R$ be a (commutative, unital) ring. Define the norm of a nonzero ideal $I$ of $R$ to be $$N(I) = \# R/I$$ which we take to be $\infty$ if the quotient ring is infinite. Also define $N(0) = 0$. I ...
0
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0answers
75 views

The $\Lambda$ function which is the summatory function of the Liouville function $\lambda$. Prove $\Lambda$ is multiplicative.

I'm trying to do part (b) of the following homework problem. I have done the other 3 parts but I'm including them for context. I just can't figure out part (b). Can you help we with it. I believe ...
1
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1answer
23 views

multiple sums including moebius mu function?

I like to prove in elementary way that $$ \sum_{k=1}^x\sum_{j=1}^k f\left(\frac jk\right)M\left(\frac xk\right)=\sum_{\nu=1}^{A(x)}f(r_{\nu}) $$ where $$ M(x)=\sum_{1\leq l\leq x} \mu(l) $$ $\mu$ ...
0
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0answers
16 views

Estimation of the convolution of two multiplicative functions

Let $f,g:\mathbb{N}\to \mathbb{C}$ be two multiplicative arithmetic functions. Assume that we know an asymptotic behavior of $f$ and $g$. Is there any general result for asymptotic behavior of the ...
-2
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1answer
35 views

function is multiplicative [closed]

Consider $ f(n) = \begin{cases} 0 & \quad \text{if } n \text{ is even}\\ 1 & \quad \text{if } n \equiv 1 \text{ mod } 4\\ -1 & \quad \text{if } n \equiv -1 ...
0
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1answer
89 views

multiplicative function - gcd - lcm [closed]

How can I show if $f$ is multiplicative and $x,y \in \mathbb{Z}_{>0}$, that this implies that $f (\text{gcd} (x,y)) * f ( \text{lcm} (x,y)) = f(x)*f(y)$? Thank you very much.
-2
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1answer
65 views

invertible element of z28 [closed]

now I find all invertible element of Z/28Z is {1,3,5,9,11,13,15,17,19,23,25,27} and mt way is listing all the element in 0 < a < 28 and each element to check their gcd is equal to 1, I think ...
11
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2answers
308 views

Let $(a_n)$ be a strictly increasing sequence of positive integers such that: $a_2 = 2$, $a_{mn} = a_m a_n$ for $m, n$ relatively prime.

Let $(a_n)$ be a strictly increasing sequence of positive integers such that: $a_2 = 2$ and $a_{mn} = a_m a_n$ for $m, n$ relatively prime. Show that $a_n = n$, for every positive integer $n$. This ...
1
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0answers
21 views

Karatsuba Multiplication Rule in dividing a Number in two parts

In Karatsuba algorithm for multiplying two numbers, we divide each number into two. For example: x= 1234 y= 2456 Then a = 12, b = 34, c = 24 , d = 56 ...
1
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1answer
60 views

Are there (restricted) instances when the deficiency and sum-of-proper-divisors functions are multiplicative?

A function $f : \mathbb{N} \rightarrow \mathbb{Q}$ is said to be multiplicative if $$f(ab) = f(a)f(b)$$ whenever $\gcd(a,b)=1$. It is known that the sum-of-divisors function $$\sigma(x) = \sum_{d \...
9
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2answers
297 views

Minimum possible value of $f(2007)$ where $f(m f(n)) = n f(m)$, $m,n\in \Bbb N$

If $f$ is from positive integers to positive integers and satisfies $f(m f(n)) = n f(m)$ then find the minimum possible value of $f(2007)$. My work so far: $f(1) = 1$ . Proof: Suppose $f(1) = k \...
2
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0answers
28 views

Closed-form for $F(n) = \sum_{d|n} \omega_x(d) $

Given $x\in \mathbb{C}$ let's define the function $$\cases { \omega_x(n)=1, \quad n=1 \\ \omega_x(n)=x^r, \quad n=p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r}}$$ a) Prove that $\omega_x(n) \colon \mathbb{...
1
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2answers
395 views

How to prove that the Legendre symbol is multiplicative?

The proof is given here in the answer Proving $(\frac{n}{p})$, a Legendre symbol, is multiplicative But I do not understand it, Also the definition in the book for Legendre symbol says that if $p|a$ ...
1
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1answer
101 views

Multiplicative functions and the sum of all divisors: $\sum_{d\mid2020}{\sigma(d)}$

Doing more practice for my final and I need some help with the following: Evaluate: $$\sum_{d\mid2020}{\sigma(d)}$$ where $\sigma(n)$ is the sum of all divisors of n. The hint given specifically ...
7
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3answers
72 views

Explain why the following is or isn't a multiplicative function.

I'm working through practice problems for my exam in class and I need some help with the following problem: A function $f(n)$ is defined to be the greatest power of $2$ that divides $n$. For example, ...
2
votes
2answers
173 views

Summing a multiplicative function

$f(n)$ is a multiplicative function, meaning $f(m\cdot n)=f(m)\cdot f(n)$. I want to evaluate the sum: $$(1)\qquad\sum_{k=1}^{n}f(m\cdot k)$$ over a fixed $m$. Because $f$ is multiplicative, I can ...
2
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1answer
60 views

How to generalize Newman's simplification of O-Tauberian theorem?

Can you prove the next theorem: Let $f$ be Dirichlet series with real, positive coefficients $(a_n>0)$. If $f$ is holomorphic on $\Re(z)\ge1$, but has one singularity at $z=1$, then $\lim_\limits{...
3
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1answer
161 views

Group automorphism of multiplicative group of real number field

Let $\mathbb{R}$ be the real number field and $\mathbb{R}^{\times}$ be the multiplicative group of it. $\mathrm{Aut}(\mathbb{R}^{\times})$ denotes the group automorphism of $\mathbb{R}^{\times}$. [...
1
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2answers
38 views

Is $n^{(p-1)/2}\equiv -1\pmod p$ implies $n$ is a primitive root modulo $p$?

Let $p$ be an odd prime, $n$ be any integer, if $$n^{(p-1)/2}\equiv -1\pmod p,$$ is it always true that $k=p-1$ is the smallest positive integer satisfy $$n^k\equiv 1\pmod p?$$ This is the little ...
1
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1answer
71 views

Proof Verification: Let $f(\chi)$ be the conductor of $\chi$, proof that $f(\chi)=f(\chi_1)\cdots f(\chi_r)$.

This is a detailed problem, let me write down the problem and the process I have done: Assume that $k=k_1k_2\cdots k_r$ where $k_i$ and $k_j$ are relatively prime for $i\neq j$. Let $\chi$ be a ...
2
votes
1answer
55 views

If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.

I'm reading the proof of the multiplicative property of $$s_k(n)=\sum_{d|(n,k)}f(d)g\bigg( \frac kd\bigg)$$ The book wrote that in order to understand the proof, we need to know if $a,b,k,m$ are ...
2
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1answer
102 views

How to multiply the radical expressions and simplify your answer?

I am trying to multiply and simplify the following radical expression. $$(\sqrt{x}+5 - 4)(\sqrt{x}+5+4)$$ According to the book, the answer is $x - 11$ However, I am confused about how this even ...
0
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1answer
25 views

If f is multiplicative, then will $f\left(\frac{a}{b}\right)$ be multiplicative for coprime $a,b$?

Given that $a$ and $b$ are coprime integers, i.e $gcd(a,b)=1$ then for any multiplicative function $f$ will $f\left(\frac{a}{b}\right)$ be multiplicative? i.e Will following property hold $$f\left(...
0
votes
1answer
104 views

Since $\zeta(1) \neq 1$, does this mean that $\zeta$ is not multiplicative?

Let $\zeta(s)$ be the Riemann zeta function, that is, $$\zeta(s) = \sum_{n=1}^{\infty}{\frac{1}{n^s}}.$$ A function $g$ is said to be multiplicative if, whenever $\gcd(x,y)=1$, we have $$g(xy) = g(x)...
-1
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2answers
46 views

$f(n) = \Sigma_{d|n} \mu(n/d)F(d)$

The question says: If $F(n) = \Sigma_{d|n} f(d)$ for every positive integer $n$, prove that $f(n) = \Sigma_{d|n} \mu(n/d)F(d)$. What I know so far is that divisors of $n$ can be paired together. ...
3
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2answers
114 views

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
votes
1answer
42 views

Cardinality set of multiples

Given an arbitrarily large set of natural numbers greater than one, S = {$p_0$, $p_1$, ... $p_n$} product of S = $\prod_{i=0}^n\ p_i$ define M as the set of all natural numbers that are multiples ...
0
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1answer
24 views

How much a variable contributed to the result of some cost function

I have this simple cost function: $\sum_{i=1}^n d_i\times h_i \times a_i$ I wanted to analyze, for example, how much the $a$ component/variable contributed to the final cost function. In other ...
2
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0answers
82 views

Asymptotic estimate for the sum $\sum_{n\leq X}\mu(n)\tau(n)$?

Just trying to figure out what would be the asymptotic relation for the expression $\sum_{n\leq X}\mu(n)\tau(n)$, where $\tau$ corresponds to the number of divisors function (often named $\sigma_0$ ...
2
votes
1answer
97 views

Modulo Arithmetic: Proof of Basic Property

If a,b,c,k $\in\mathbb{Z\cap{(N\cup{0}})}$ and a$\equiv$b(mod c) then prove that a$^{k}\equiv$b$^{k}$(mod c). I know how to prove it using induction but I wanted to know if there is a method that only ...
3
votes
3answers
98 views

If $S(n)$ is an odd integer, what is the sum of all possible $\frac1n?$ [closed]

If $n$ is a positive integer, let $S(n)$ be the sum of all the positive divisors of $n$. If $S(n)$ is an odd integer, what is the sum of all possible $\frac1n?$ The function $S$ is multiplicative ...
1
vote
1answer
45 views

Word to describe factor of x or 1/x

There is a well known video of a helicopter with 5 evenly spaced rotor blades where the rotor is synced with a digital camera's frame rate. Assuming the camera is at 60 hertz (60 frames per second), ...
1
vote
2answers
75 views

Is there an analytic proof of change of bases in logarithms?

Usually change of bases in logarithms is just observance $$x=b^{\log_bx}\implies \log_kx={(\log_bx)}{(\log_kb)}\implies \log_kb=\frac{\log_kx}{\log_bx}.$$ Supposing apriori we do not know inverse of ...
1
vote
1answer
58 views

Upper bound on coefficients of the logarithmic derivative of a certain Dirichlet series

For a multiplicative arithmetic function $f(n)$, we define $ F(s) = \sum_{n\ge1}^{} \dfrac{f(n)}{n^s}$. We then define the coefficients $\Lambda_f (n)$ by $$ -\dfrac{F'(s)}{F(s)} = \sum_{n\ge1}^{} \...
1
vote
1answer
63 views

Landau notation and a preliminary step in the computation of the average order of $\sigma(n)$

My question is to show that for all real numbers $x \ge 2$: $$\sum_{n \le x} \frac {\sigma (n)}{n} = \frac {\pi ^2}{6}x + O(\log x)$$ I think the first step is to break down the sigma function: $$\...

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