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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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 answers
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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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Discontinuous multiplicative Linear Functional on a non complete normed Algebra

I have come across the following proposition in the book "Complete Normed Algebras" by F. F. Bonsall and J. Duncan in section 16 on page. Definition: A multiplicative linear functional on ...
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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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1 answer
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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 ...
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5 answers
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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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2 votes
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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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2 answers
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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 ...
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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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1 vote
0 answers
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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 ...
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2.20 Question Introduction to analytic number theory

The following question is on page 48 apostol Introduction to analytic number theory( 20) Let $P(n)$ be the product of the positive integers which are less than equal to $n$ and relatively prime to $n$...
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A question in solution of a problem involving multiplicative functions

This question is question 2.3 of Apostol Introduction to analytic number theory. It's solution image: My question: how did in RHS on last line of solution author got $n / \phi(n) $ as $n = p_{1}^{a_1}...
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5 votes
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Efficient summation of multiplicative functions on intervals

Suppose that $f: \mathbb{N} \to \mathbb{C}$ is a multiplicative function; i.e., $f(mn) = f(m)f(n)$ if $m, n$ are coprime. I'm interested in efficiently calculating sums of the form $\sum_{k=1}^N f(k)$...
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Is $\frac{\sigma(n)}{{2n}}$ submultiplicative or supermultiplicative?

This question is an offshoot of this earlier post. It is known that the abundancy index $$I(n) = \frac{\sigma(n)}{n}$$ is a multiplicative function. (Note that the divisor sum $\sigma$ is also ...
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2 votes
1 answer
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Is this function multiplicative and if so what is its value at prime powers?

For odd numbers $n$ let: $$a(n) = \sum_{d^2|n} d \frac{\sigma^*(n/d^2)}{2^{\omega(n/d^2)}}$$ where $\sigma^*(k) = $ sum of unitary ($\gcd(d,k/d)=1$) divisors of $k$ and $\omega$ counts the prime ...
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2 answers
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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 ...
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2 votes
3 answers
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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^...
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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 ...
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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.
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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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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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3 votes
1 answer
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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, ...
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4 votes
1 answer
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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+\...
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1 vote
1 answer
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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 ...
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