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Questions tagged [multiplicative-function]

In number theory, a multiplicative function is a function defined on positive integers such that f(ab)=f(a)f(b) for a,b coprime. E.g. Euler's totient function, sum of divisors and number of divisors are multiplicative functions.

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Multiplicative function evaluated at prime decompositions

Say $f$ is multiplicative and $m = p^{\alpha}q^{\beta}$ where $p,q$ are prime. Then do we have $f(m) = f(p^{\alpha})f(q^{\beta})$? If so why?. I know this hold when the powers are 1 but have not been ...
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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 ...
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33 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{...
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1answer
52 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 $\mathbb{R}^{\times}$. [...
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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 ...
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23 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 ...
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42 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 ...
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1answer
65 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 ...
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23 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(...
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88 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)...
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$f(n) = \Sigma_{d|n} \mu(n/d)F(d)$

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. ...
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63 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 ...
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1answer
22 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 ...
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23 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 ...
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63 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 $\sigma_0$ ...
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1answer
64 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 ...
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87 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 ...
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42 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), ...
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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 ...
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48 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}^{} \...
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62 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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79 views

If $f$ is an arithmetic function with $f(1)=1$ then $f$ is multiplicative

I'm studying analytic number theory for undergraduates and I read this theorem in Tom Apostol's book on the second chapter: Theorem 2.12. If $f$ is multiplicative then $f(1)=1$ And under need ...
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1answer
25 views

$g(m)$ is a multiplicative fuction.

$g(m)=\sum_{n=1, (n,m)=1}^{n=m} e^{2\pi i n /m}$ I tried but didnt able to show it. Also how to show that $g(p)=-1$ for $p$ prime. I tried it just by splitting the series and collecting the terms ...
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27 views

How to multiply two functions with two variables and manually build a plot

I am trying to learn how to work with functions and I have some things that I didn't fully understand. How do I multiply and plot a function that is the result of a multiplication of two other ...
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1answer
49 views

Proving that two multiplicative functions are equal

I need to prove the following statement: "Let 'f' and 'g' be two multiplicative functions such that f(pk) = g(pk) for each prime p and k $\geqslant$ 1. Prove that f = g" I have tried to approach ...
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55 views

Help in showing that a function is multiplicative

I am solving this same very question: For any positive integer $n$, show that $\sum_{d|n}\sigma(d) = \sum_{d|n}(n/d)\tau(d)$ I want to approach this question via proving the multiplicativity of the ...
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Some convincing reasoning to show that to prove that Ramanujan tau function is multiplicative is very difficult

I am curious to know (some words or reasoning about how to justify) why to prove that the so-called Ramanujan $\tau$ is a multiplicative function is (was) very difficult. In this Wikipedia is showed ...
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how do I solve this, $\tau(m) = 21$ [closed]

I know how to do this for an even number, but I don't understand how I would do it for an odd number, $\tau (m) = 21$.
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199 views

Order of operations for multiplying three matrices

If I have a $1\times 2$ matrix $A$, a $2\times 2$ matrix $B$, and a $2\times 2$ matrix $C$, and am asked to calculate ABC, is there a specific order in which I have to carry out the multiplication? ...
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48 views

How to calculate the multiplicator in a sum like sum += sum*(mulu^n)

In a section of a personal PHP project, I would like to calculate the spending factor in a rule where we spend Nth time the previous payment done. Here is an example of spending. ...
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198 views

How to do a sum of a function over all divisors of an integer with Maple?

I'd like to define sumdiv in Maple such that this: with(numtheory); f:=x->x^2; sumdiv(f(d)*mobius(100/d), d=1..100); would ...
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In the definition of the Mobius function and basic properties, can we change square-free to something else?

The Mobius function isolates square-free numbers ie numbers. Can we do this with other numbers with certain properties?
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Lower bound for sum of a multiplicative function based on lower bound on value of primes

Let $f$ be a non-negative multiplicative function. Suppose we have some bound on $\sum_{p \le x} f(p)$, where the summation is over primes. Is it possible to give a lower bound on $\sum_{n \le x} f(n)$...
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31 views

Increment in multiplying

I want to calculate the total cost of an investment in a game. For each level, the costs is increased with 5000, i.e. lvl 1 costs 5000, lvl 2 costs 1000 and lvl 3 costs 1500 etc. At level 3, the ...
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Does the Euler product stand for $a(n)=rad(n)$?

Does the Euler product stand for $a(n)=rad(n)$? Or more generally, for multiplicative functions which are not completely multiplicative? Where rad is the product of a number's distinct prime factors....
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68 views

On arithmetic functions whose Dirichlet series has a special kind of abscissa of absolute convergence

For a function $f: \mathbb N \to \mathbb C$ , let $\sigma_c(f) , \sigma_a(f)$ denote the abscissa of convergence and the abscissa of absolute convergence respectively of the Dirichlet series $\sum_{n=...
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find strictly increasing function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that $f(2)=2$ and $f(x\cdot y)=f(x)\cdot f(y)$ when $\gcd(x,y)=1$

The problem is as in title... I understand that, according to for example answers to this question, trivial solution is $f(x)=x$. However, I don't see is weakening condition $f(x\cdot y)=f(x)\cdot f(y)...
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1answer
96 views

Show that $\sigma(n)$ is equally often even and odd

Let $\sigma(n)$ be the divisor sum of $n$: $$ \sigma(n) = \sum_{d|n} d. $$ I was interested in the parity of $\sigma(n)$ and tried to check whether $\sigma(n)$ is unexpected often even for odd $n$ ...
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Interesting multiplicative Ramanujan-like q-expansions

We all know the full modular (cusp) form of weight 12 $$ \Delta(z) = \sum_{n=1} \tau(n)q^n = q \prod_{n=1} (1-q^n)^{24} $$ that generates the multiplicative Ramunujan tau function $\tau(n)$. Today I ...
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118 views

When is the product of two Dirichlet $L$-functions also a Dirichlet $L$-function?

This question came up while trying to learn about the Riemann zeta function - as is the guilty pleasure of many of us. A Dirichlet $L$-function is notated as $$L\{z,\chi\}(z) = \sum_{n=0}^{\infty}\...
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Always true : $ ( \sum_{i=1}^d x_i^a )( \sum_{i=1}^d y_i^a ) = ( \sum_{i=1}^d z_i^a )$

All variables are integers $>-1$. Consider $ f(a) = d $ such that d is the smallest value $>1$ such that : It is always true that $$ ( \sum_{i=1}^d x_i^a )( \sum_{i=1}^d y_i^a ) = ( \sum_{i=1}...
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1answer
106 views

How to reformulate a multiplicative formula (with two primes, perhaps like totient-function)?

In the work on another question in MSE I have a formula $f(n)$ whose pattern depending on $n \in \Bbb N$ I want decode into an algebraical formula (see a short rationale of $f(n) at the end). ...
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1answer
153 views

Find all $n$ such $\sigma(n) = 546$

Find all $n$ such $\sigma(n) = 546$. I find $n = 180$ is an answer. But I know there is more. I tried to used the formula $$\sigma(n) = \frac{{p_1}^{a_1 + 1} - 1}{p_1 - 1} \ldots \frac{{p_k}^{a_k + 1} ...
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908 views

How do you get the multiplicative inverse by reading a multiplication table of modulo $n$?

Here is the multiplication table for$\mod 7$: https://gyazo.com/82fb45bd89f61df3b44f00f67efc63c1 How do I read this to get the multiplicative inverse of something, for example like: $5\mod 7$ I ...
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264 views

General summations of multiplicative functions

I have read about multiplicative functions. I also came across summations involving summation of multiplicative functions as well. Some summations were only over divisors of a number, which can be ...
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1answer
261 views

Euler's Totient Function as a multiplication of terms

I remember once reading an interesting, simple way to prove, that $φ(a⋅b) = φ(a) φ(b)$ iff $a$ and $b$ are coprime. The proof started with the assumption that $φ(n) = n(1−1/p_1)(1−1/p_2)…(1−1/p_r)$ ...
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1answer
49 views

Wrong millimeter values for standard paper sizes at Wikipedia?

On the Wikipedia page https://en.wikipedia.org/wiki/Paper_size there are tables for standard paper sizes. The table "ANSI and CAN paper sizes" has paper size values expressed in ...
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50 views

A strange multiplicative function

I have a function that looks like $$f_n(i) = \max\left(\left\lfloor \frac{i-n}{2} \right \rfloor + 1, 0\right)$$ and I would like to write it as a nice arithmetic function. To give an idea, another ...
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2answers
87 views

Sum of divisors, a congruence

Let $\sigma_r(n)=\sum_{d|n}d^r$ where the sum is over all the integers $d=1,\dots,n$ which divide $n$. I am conjecturing $$\sum_{m=1}^{p-1}m^2 \sigma_3(m)\sigma_3(p-m)\not\equiv 0\pmod{p^2}$$ for ...
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105 views

Multiplicative Functions and Totient Function

I have two questions. If $f$(n) is a multiplicative function defined on the positive integers, is $g(n)=$$\frac{f(n)}{n}$ multiplicative as well? I think the answer is yes, but I don't know how ...