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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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}^{\...
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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 ...
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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 ...
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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" ...
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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 ...
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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 ...
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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 ...
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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 $m$-...
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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 ...
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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
...
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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 ...
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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 ...
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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 $\...
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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:
$$
(...
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If $b|a$ and $f$ is a completely multiplicative function, how can I show that $f(a/b) = f(a) / f(b)$?
I know that $f(ab) = f(a)f(b)$ , but I'm not quite sure how to use this information.
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Full derivation inside of twin prime statement in terms of multiplicative arithmetic functions. How can the last formula be rearranged?
Let $(\cdot\mid\cdot) : \Bbb{N}\times\Bbb{N} \to \Bbb{Z}_2$ be the divisibility function which takes on the value $(x|y) = 1$ whenever $x$ divides $y$ and the value $(x|y) = 0$ whenever it does not ...
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Continuous Factorial
I am working on a theory of quantum information and am unsure on some of the mathematical formalism I need.
I have learned that integration can be thought of as summing up infinitely thin slices.
My ...
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Summatory function of Euler-phi
Let $F(n) = \sum_{d^2|n} \phi(d)$. We must show that if $F(1) = 1$, and if $n>1$ factors as $n=p^{a_1}_1p^{a_2}_2...p^{a_m}_m$, then
$$
F(n)=\prod_{i=1}^{m} p^{[a_i/2]}_i.
$$
If I understood ...
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Why did we stop creating functions at exponents?
It seems that every new function is just a function that asks something of the one before it, for example: First we have addition. That is the starting point. Now, multiplication, which asks "How ...
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A question about the property of completely multiplicative functions
I am self-studying number theory. In Apostol's number theory textbook, Theorem 2.17 states:
Let $f$ be multiplicative. Then $f$ is completely multiplicative if
and only if $$f^{-1}(n)=\mu(n) f(n) \...
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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, ...
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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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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 ...
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3
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213
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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 ...
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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 $n=1$ ...
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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 $f:\mathbb N \to\mathbb N$ ...
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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\...
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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 ...
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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 $a$ ...
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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 ...
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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
$$\mu(n)= \left\{ \...
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totally multiplicative function
Let $f$ be a totally multiplicative function, $f(n+N)=f(n)$ for every $n \in \mathbb{N}$. Prove that $|f(n)|=1$ for every $n \in \mathbb{N}$, where $gcd(n,N)=1$
My approach is to prime factorise $n+N=\...
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Equivalence between convex combination and multiplicative form
I am interested in understanding the relationship between convex combination and a multiplicative form.
Let $x,y > 0$. Let the convex combination be $f(\gamma) = \gamma x + (1-\gamma) y$ where $\...
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When multiplying two or more terms in an integral, can I calculate the terms seperately and then multiply?
For example, say I have an equation that is $\int_0^\infty x g(x)f(x)dx$
Can I calculate each term seperately, say for example my maximum $x$ value is 500.
Is it something like $500\int_0^{500} g(x)\...
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Sums of divisors of a Jacobi symbol
$$f(n)= \genfrac(){}{0}{-15}{n}$$ if $n \neq 1$ is odd.
What is $\sum_{d \mid n}f(d)$ ?
Here is what I tried : Let $n=p_1^{a_1} \cdot \cdot \cdot p_r^{a_r}$, for $p_1,...,p_r$ odd primes. If $p_1,...,...
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Euclidean algorithm and Multiplicative inverse in RSA Cryptosystem
I understand RSA Cryptosystem, the Euclidean algorithm, and mod, however, I can't seem to understand how to solve the following problem.
-Use Euclidean algorithm to compute the multiplicative inverse $...
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Prove that $G(n) = \sum_{d|n} \frac{n}{d}\tau(d)$ is multiplicative.
I don't know how to prove this because it seems to me that $f(d)=\frac{n}{d}\tau(d)$ is not multiplicative.
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Sum of an arithmetic function over the divisors of an integer is multiplicative
I am trying to prove the following claim:
Let $f$ be an arithmetic function and, for $n \in \mathbb{Z}$ with $n > 0$, let
$$
F(n) = \sum_{d\ \vert\ n,\ d > 0}f(d)
$$
If $F$ is multiplicative, ...
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Group operator vs. complex number multiplication operator in the multiplicative property of a character of a group [closed]
A character $f$ of a group $G$ is defined as a complex-valued function defined on $G$ that has the multiplicative property $f(ab) = f(a)f(b)$ for all $a, b$ in $G$, and if $f(c) \ne 0$ for some $c$ in ...
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Let $f$, $g$ be multiplicative functions, not identically $0$ such that $f(p^k)=g(p^k)$ for each prime $p$ and $k \ge 1$. Prove that $f = g$.
Let $f$ and $g$ be multiplicative functions that are not identically $0$ and such that $f(p^k)=g(p^k)$ for each prime $p$ and $k\ge1$. Prove that $f=g$.
Source: Elementary Number Theory by David M. ...
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The Question about Euler's Totient Function : How phi(i*p) = p*phi(i)? Here, p is prime and p divides i, phi-> Euler totient function
I have a question about the Euler totient function. I am new to the number theory and i don't know where to start to prove this. If $p$ is prime and $p$ divides $i$,
$$ Φ (i\cdot p) = p \cdot Φ (i),$$...
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derivate is based on addition, is there a muliplication analogon?
like
$$
\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
$$
does it ...
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Dirichlet convolution of a multiplicative function with itself
Let $f$ be a multiplicative function, i.e. for all coprime $a,b\in\mathbb{N}\quad f(ab)=f(a)f(b)$.
Consider:
$$(f*f)(n)=\sum_{d\vert n}f(d)f\left(\frac{n}{d}\right),\quad\text{where * is Dirichlet ...
user764256
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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 ...
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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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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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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 ...
user125368
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Multiplicative Functions Proof
I don't know if this question was asked before, but I could not find it on this forum: a function $f(n)$ is multiplicative if $f(mn)=f(m)f(n)$, where m and n are coprime and positive. If $d(n)$ is the ...
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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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Any results about auto-correlation function of Mertens function
One can define $f(x) = M(e^x)/\sqrt{e^x}$, where $M$ is Mertens function. It looks like some sort of stationary random process (yes, I know it's not random process, see plot below), namely it lives ...
