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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Probability that one random number among many has a unique prime factor

If I sample $N+1$ integers $x, x_1, \ldots, x_N$ uniformly and independently from $\{1, \ldots, M=2^k\}$, what is the probability that $x$ contains a prime divisor that does not divide any of the $\{...
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Average of Multiplicative Function

Let $\mathbb{S}^1$ be the unit circle. Suppose $g:\mathbb{N}\to\mathbb{S}^1$ is a completely multiplicative function such that $$\sum_{\substack{p\in\mathbb{P}\\ g(p)\neq 1}}\frac{1}{p}<\infty.$$ ...
defacto's user avatar
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Let $F: \mathbb R^n \to \mathbb R^m$ be smooth and such that for all $a \in \mathbb R$, $x \in \mathbb R^n$, $F(ax) = aF(x)$. Prove $F$ is linear

Let $F: \mathbb R^n \to \mathbb R^m$ be smooth and such that for all $a \in \mathbb R$, $x \in \mathbb R^n$, $F(ax) = aF(x)$. Prove $F$ is linear. I am sorry to say this, but I am really stuck on ...
Squirrel-Power's user avatar
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Summation notation over divisors confusion

What does the following summation notation represent? $\sum\limits_{d_1 \mid a, \; d_2\mid b}f(d_1d_2)=\sum\limits_{d_1\mid a }\sum\limits_{d_2 \mid b}f(d_1)f(d_1)=\sum\limits_{d_1\mid a}f(d_1)\sum\...
Jason Xu's user avatar
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Linearization of Multiplicative model

I've read a paper about Huff model, and I have a question for linearization technique of Multiplicative model. How does following linearization work? $U_{ij} = X_{1j}^\alpha X_{2j}^\beta X_{1j}^{\...
최용훈's user avatar
1 vote
1 answer
75 views

Prove that there are infinitely many natural number such that $σ(n)>100n$

The problem is as follows: Prove that there are infinitely many natural numbers such that $σ(n)>100n$. $σ(n)$ is the sum of all natural divisors of $n$ (e.g. $σ(6)=1+2+3+6=12$). I have come to the ...
Mathology's user avatar
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Euler's Totient Function - Multiplicativity via Group of Units

The ring $\mathbb{Z}_{n m}$ is isomorphic to the ring $\mathbb{Z}_n \times \mathbb{Z}_m$ if $\operatorname{gcd}(m, n)=1$ holds. Let us prove the property that the Totient Function $\varphi$ is ...
calculatormathematical's user avatar
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1 answer
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Sum of multiplicative functions

Let $f_1, f_2$ multiplicative functions such that $f_1 \ne 0$ and $f_2 \ne 0$. I want to show that $f_1 + f_2$ is multiplicative if and only if $f_1 = -f_2$ I tried starting the "only if" ...
Tamir Vered's user avatar
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2 answers
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When is the norm of a ring $\Bbb Z[\sqrt{-p}]$ multiplicative?

I got inspired by quadratic rings and zeta functions. I know for instance that the norm $a^2 + 17 b^2$ for the ring $\Bbb Z[\sqrt{-17}]$ is multiplicative yet the ring $\Bbb Z[\sqrt{-17}]$ is not a ...
mick's user avatar
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On a conjecture involving multiplicative functions and the integers $1836$ and $137$

We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see the corresponding ...
user759001's user avatar
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On a conjecture involving multiplicative functions and the integers $1836$ and $136$

We denote the Euler's totient as $\varphi(x)$, the Dedekind psi function as $\psi(x)$ and the sum of divisors function as $\sigma(x)$. Are well-known arithmetic functions, see Wikipedia. I would like ...
user759001's user avatar
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Multiplicative complex function has mean value

I came up with the following question, and I don't have proof of it. Let $m>1$ be a positive integer. Let $f:\mathbb{N}\to \mathbb{C}$ a multiplicative function, whose image is a subset of the $m$-...
AMarchionna's user avatar
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105 views

Examples of quasi-logarithmic functions on natural numbers.

I saw this question recently and was curious about the value of the smallest symmetric group $S_k$ that has an element of order $n$. Call $k$, which depends on $n$, $f(n)$. I checked a few small ...
Greg Nisbet's user avatar
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The mean square of $d_k(n)$

Let $d_2(n)=d(n)$ be the divisor function, and let $$d_k(n)=\sum_{d_1\cdots d_k= n}1=\sum_{m\cdot l= n}d_{k-1}(m).$$ Can anyone point me to a reference to the size of the error term when approximating ...
tomos's user avatar
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Polynomial Approximation of Multiplicative Function

Motivation: I am a undergrad studying number theory and all the multiplicative functions I studied in the course note has a polynomial upper bound, leading me to inquire where this holds for all ...
JZhou's user avatar
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Number of ways to write $2n-1$ as $4x+2y+4xy+1$ where $x$ and $y$ are nonnegative integers

I see a claim in OEIS/A1227 which state that "Number of ways to write $2n-1$ as $4x+2y+4xy+1$ where $x$ and $y$ are nonnegative integers equals number of odd divisors of $n$." I can't see ...
Kevin's user avatar
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Name of conjecture about correlation of $\lambda(n)$ and $\lambda(n+1)$

I remember reading about a conjecture one night a while ago, but I can’t seem to find anything about it anymore. I have forgotten it’s name, but I believe the conjecture went as follows: Suppose $\...
Snacc's user avatar
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Determining all arithmetic multiplicative functions that are idempotent to the convolution product

Exercise. Determine all the arithmetic multiplicative functions that are idempontent to the convolution product, i.e., determine all the functions $f$ such that, for every $a \in \Bbb N$, we have: $$ (...
xyz's user avatar
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If $b|a$ and $f$ is a completely multiplicative function, how can I show that $f(a/b) = f(a) / f(b)$?

I know that $f(ab) = f(a)f(b)$ , but I'm not quite sure how to use this information.
user1063905's user avatar
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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 ...
Daniel Donnelly's user avatar
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1 answer
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Continuous Factorial

I am working on a theory of quantum information and am unsure on some of the mathematical formalism I need. I have learned that integration can be thought of as summing up infinitely thin slices. My ...
moboDawn_φ's user avatar
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48 views

Summatory function of Euler-phi

Let $F(n) = \sum_{d^2|n} \phi(d)$. We must show that if $F(1) = 1$, and if $n>1$ factors as $n=p^{a_1}_1p^{a_2}_2...p^{a_m}_m$, then $$ F(n)=\prod_{i=1}^{m} p^{[a_i/2]}_i. $$ If I understood ...
DrJimmour's user avatar
2 votes
0 answers
90 views

Why did we stop creating functions at exponents?

It seems that every new function is just a function that asks something of the one before it, for example: First we have addition. That is the starting point. Now, multiplication, which asks "How ...
Joel Castro's user avatar
2 votes
1 answer
300 views

A question about the property of completely multiplicative functions

I am self-studying number theory. In Apostol's number theory textbook, Theorem 2.17 states: Let $f$ be multiplicative. Then $f$ is completely multiplicative if and only if $$f^{-1}(n)=\mu(n) f(n) \...
Josh's user avatar
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Necessary and sufficient condition to be completely multiplicative

I want to prove that $f*f=f \tau$ iff $f$ is completely multiplicative. The "if" part was relatively easy, using $f(g*h)=(fg)*(fh)$ and plug $g=h=1$ for all $n$. Juxtaposition is ordinary, ...
mathhello's user avatar
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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 ...
MathBS's user avatar
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Fast consecutive prime multiplication

Are there any fast algorithms for multiplying consecutive prime numbers? I found this question about a pattern when multiplying consecutive primes. I was wondering if there is a multiplication ...
murage kibicho's user avatar
3 votes
3 answers
236 views

Why is it important that the $p$-adic absolute value satisfy multiplicativity?

I suppose my question is really "why do we require norms in general to satisfy multiplicativity?". I ask this because for the usual absolute value on $\mathbb R$, I never feel like ...
D.R.'s user avatar
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-2 votes
1 answer
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$f$ is multiplicative $\implies f^{-1}$ is multiplicative. [closed]

Let $f$ be a multiplicative function i.e. $f(mn)=f(m)f(n)$ for all $m,n$ satisfying $\gcd(m,n)=1$ and $f\not\equiv 0$. Define $f^{-1}$ to be the function $g$ such that $f*g=I$ where $I(n)=1$ if $n=1$ ...
Kishalay Sarkar's user avatar
1 vote
1 answer
121 views

Find the values of $f(2)$ for which $f$ cannot be a strictly increasing and completely multiplicative function

I was solving some problems on functional equations especially on multiplicative functions today. I noticed that a strictly increasing completely multiplicative function $f:\mathbb N \to\mathbb N$ ...
Oshawott's user avatar
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7 votes
2 answers
393 views

Prove that $f(n)=n^2$ where $f$ is a strictly increasing multiplicative function with $f(2)=4$.

Let $f:\mathbb N\to\mathbb N$ be a strictly increasing function with $f(2)=4$ which is completely multiplicative i.e $f(ab)=f(a)f(b)$ for all $a,b\in\mathbb N$. Prove that $f(n)=n^2$ for all $n\in\...
Oshawott's user avatar
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2 votes
1 answer
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How to prove that a function on a quotient set $\mathbb{Z}/n\mathbb{Z}$ is well defined?

I am asked to prove that a function under a quotient set is well-defined. I know that for a function to be well-defined, two mappings or images found in the co-domain/range can't be mapped from the ...
Khalil Alashy's user avatar
2 votes
1 answer
85 views

Factorials and Place Value

I recently came across this question from a non-calculator exercise. The units and tens place value digits I can see as $0$ and $0$ since in $12!$ we have $10*5*2=100$ but is there a way to find $a$ ...
Jon Percival's user avatar
2 votes
1 answer
221 views

How would one motivate/know to introduce the Dirichlet character in the formula for the number of lattice points on a circle of radius $\sqrt N$

Grant's masterful video https://www.youtube.com/watch?v=NaL_Cb42WyY&ab_channel=3Blue1Brown ("Pi hiding in prime regularities") describes a way of computing $\pi$ that ultimately leads us ...
D.R.'s user avatar
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0 votes
1 answer
109 views

Do these multiplicative functions exist?

I read about the multiplicative function $\mu$ defined by Möbius. Which is for any given $n\in \mathbb{N}$ such as $n=p_1^{\alpha_1}\cdot ....\cdot p_r^{\alpha_r}$ is defined as $$\mu(n)= \left\{ \...
David González's user avatar
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0 answers
34 views

totally multiplicative function

Let $f$ be a totally multiplicative function, $f(n+N)=f(n)$ for every $n \in \mathbb{N}$. Prove that $|f(n)|=1$ for every $n \in \mathbb{N}$, where $gcd(n,N)=1$ My approach is to prime factorise $n+N=\...
popping900's user avatar
0 votes
1 answer
71 views

Equivalence between convex combination and multiplicative form

I am interested in understanding the relationship between convex combination and a multiplicative form. Let $x,y > 0$. Let the convex combination be $f(\gamma) = \gamma x + (1-\gamma) y$ where $\...
Froozle's user avatar
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2 votes
1 answer
346 views

When multiplying two or more terms in an integral, can I calculate the terms seperately and then multiply?

For example, say I have an equation that is $\int_0^\infty x g(x)f(x)dx$ Can I calculate each term seperately, say for example my maximum $x$ value is 500. Is it something like $500\int_0^{500} g(x)\...
OpenSauce's user avatar
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3 votes
1 answer
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Sums of divisors of a Jacobi symbol

$$f(n)= \genfrac(){}{0}{-15}{n}$$ if $n \neq 1$ is odd. What is $\sum_{d \mid n}f(d)$ ? Here is what I tried : Let $n=p_1^{a_1} \cdot \cdot \cdot p_r^{a_r}$, for $p_1,...,p_r$ odd primes. If $p_1,...,...
popping900's user avatar
-1 votes
2 answers
156 views

Euclidean algorithm and Multiplicative inverse in RSA Cryptosystem

I understand RSA Cryptosystem, the Euclidean algorithm, and mod, however, I can't seem to understand how to solve the following problem. -Use Euclidean algorithm to compute the multiplicative inverse $...
MaïkaJ.'s user avatar
-1 votes
1 answer
67 views

Prove that $G(n) = \sum_{d|n} \frac{n}{d}\tau(d)$ is multiplicative.

I don't know how to prove this because it seems to me that $f(d)=\frac{n}{d}\tau(d)$ is not multiplicative.
Brett Lane's user avatar
1 vote
1 answer
166 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, ...
Ajay Tatachar's user avatar
0 votes
1 answer
64 views

Group operator vs. complex number multiplication operator in the multiplicative property of a character of a group [closed]

A character $f$ of a group $G$ is defined as a complex-valued function defined on $G$ that has the multiplicative property $f(ab) = f(a)f(b)$ for all $a, b$ in $G$, and if $f(c) \ne 0$ for some $c$ in ...
Lone Learner's user avatar
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0 votes
3 answers
715 views

Let $f$, $g$ be multiplicative functions, not identically $0$ such that $f(p^k)=g(p^k)$ for each prime $p$ and $k \ge 1$. Prove that $f = g$.

Let $f$ and $g$ be multiplicative functions that are not identically $0$ and such that $f(p^k)=g(p^k)$ for each prime $p$ and $k\ge1$. Prove that $f=g$. Source: Elementary Number Theory by David M. ...
arnav_de's user avatar
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2 votes
2 answers
84 views

The Question about Euler's Totient Function : How phi(i*p) = p*phi(i)? Here, p is prime and p divides i, phi-> Euler totient function

I have a question about the Euler totient function. I am new to the number theory and i don't know where to start to prove this. If $p$ is prime and $p$ divides $i$, $$ Φ (i\cdot p) = p \cdot Φ (i),$$...
rajat garg's user avatar
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1 answer
46 views

derivate is based on addition, is there a muliplication analogon?

like $$ \operatorname{f^o}(x) = \lim_{h\to 1} \frac{f(x*h)}{f(x)} $$ $$ \operatorname{f}(x)=e^x $$ $$ \operatorname{f^o}(x) = \lim_{h \to 1} e^{x*h}/e^{x} = \lim_{h\to 1} e^{x*h-x}=e^0=1 $$ does it ...
Philipp Ziehe's user avatar
1 vote
0 answers
184 views

Dirichlet convolution of a multiplicative function with itself

Let $f$ be a multiplicative function, i.e. for all coprime $a,b\in\mathbb{N}\quad f(ab)=f(a)f(b)$. Consider: $$(f*f)(n)=\sum_{d\vert n}f(d)f\left(\frac{n}{d}\right),\quad\text{where * is Dirichlet ...
user avatar
2 votes
5 answers
649 views

Find the highest natural number which is divisible by $30$ and has exactly $30$ different positive divisors.

Find the highest natural number which is divisible by $30$ and have exactly $30$ different positive divisors. What I Tried: I am not sure about any specific approach to this problem. Of course as the ...
Anonymous's user avatar
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2 votes
1 answer
247 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 ...
mathemagician99's user avatar
-2 votes
2 answers
429 views

RSA: what is $k$ in the formula to calculate $d$? [closed]

In RSA, to calculate $d$, when given $\phi(n)$ and $e$, I stumbled upon this formula: $$d = \dfrac{k \phi(n) + 1}{e}$$ But what does $k$ stand for? How to obtain the value for $k$? Thank you, Chris
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