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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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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1answer
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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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1answer
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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
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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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1answer
21 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
$$f\left(...
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Since $\zeta(1) \neq 1$, does this mean that $\zeta$ is not multiplicative?
Let $\zeta(s)$ be the Riemann zeta function, that is,
$$\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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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
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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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1answer
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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0answers
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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
58 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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3answers
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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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1answer
41 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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2answers
71 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 ...
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1answer
36 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
$$ -\dfrac{F'(s)}{F(s)} = \sum_{n\ge1}^{} \...
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1answer
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Landau notation and a preliminary step in the computation of the average order of $\sigma(n)$
My question is to show that for all real numbers $x \ge 2$:
$$\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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2answers
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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
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$g(m)$ 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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1answer
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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
46 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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1answer
47 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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3answers
342 views
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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1answer
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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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2answers
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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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3answers
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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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3answers
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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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0answers
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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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2answers
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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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1answer
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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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0answers
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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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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
94 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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0answers
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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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1answer
107 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
105 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
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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 $$\sigma(n) = \frac{{p_1}^{a_1 + 1} - 1}{p_1 - 1} \ldots \frac{{p_k}^{a_k + 1} ...
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1answer
812 views
How do you get the multiplicative inverse by reading a multiplication table of modulo $n$?
Here is the multiplication table for$\mod 7$: https://gyazo.com/82fb45bd89f61df3b44f00f67efc63c1
How do I read this to get the multiplicative inverse of something, for example like:
$5\mod 7$
I ...
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0answers
228 views
General summations of multiplicative functions
I have read about multiplicative functions. I also came across summations involving summation of multiplicative functions as well. Some summations were only over divisors of a number, which can be ...
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1answer
215 views
Euler's Totient Function as a multiplication of terms
I remember once reading an interesting, simple way to prove, that $φ(a⋅b) = φ(a) φ(b)$ iff $a$ and $b$ are coprime.
The proof started with the assumption that
$φ(n) = n(1−1/p_1)(1−1/p_2)…(1−1/p_r)$ ...
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1answer
48 views
Wrong millimeter values for standard paper sizes at Wikipedia?
On the Wikipedia page https://en.wikipedia.org/wiki/Paper_size there are tables for standard paper sizes. The table "ANSI and CAN paper sizes" has paper size values expressed in ...
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0answers
49 views
A strange multiplicative function
I have a function that looks like
$$f_n(i) = \max\left(\left\lfloor \frac{i-n}{2} \right \rfloor + 1, 0\right)$$
and I would like to write it as a nice arithmetic function. To give an idea, another ...
2
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2answers
81 views
Sum of divisors, a congruence
Let $\sigma_r(n)=\sum_{d|n}d^r$ where the sum is over all the integers $d=1,\dots,n$ which divide $n$. I am conjecturing
$$\sum_{m=1}^{p-1}m^2 \sigma_3(m)\sigma_3(p-m)\not\equiv 0\pmod{p^2}$$
for ...
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2answers
93 views
Multiplicative Functions and Totient Function
I have two questions.
If $f$(n) is a multiplicative function defined on the positive integers, is
$g(n)=$$\frac{f(n)}{n}$ multiplicative as well?
I think the answer is yes, but I don't know how ...
2
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1answer
82 views
Does $\sigma(\frac{m}{n})=\frac{\sigma(m)}{\sigma(n)}$ where $\sigma$ is the sum of divisor function?
My question is the same as with the title. Kindly help me prove the statement below if it is true.
What I tried: For specific value of $m$ and $n$ the equality seems to hold.
Does $\sigma(\frac{m}{...
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1answer
56 views
Two Dirichlet characters $\chi$, $\chi'$ are equal if $\chi(p) = \chi'(p)$ for almost all primes
I want to prove the following: Let $\chi, \chi'$ be two primitive Dirichlet-characters of conductor $N$. Suppose that $\chi(p) = \chi'(p)$ for all but a finite number of primes $p$. Then $\chi = \chi'$...
4
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0answers
143 views
Minimal value of summatory function of completely multiplicative functions taking values -1 and 1
Here is a very nice paper http://www.ams.org/journals/tran/2010-362-12/S0002-9947-2010-05235-3/S0002-9947-2010-05235-3.pdf which led me to thinking about the problems below.
Define the Liouville ...
