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Questions tagged [multiplicative-function]

This tag is for questions relating to multiplicative functions which are arise most commonly in the field of number theory.

43 questions with no upvoted or accepted answers
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4
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147 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 ...
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0answers
386 views

Group-theoretic proof that an increasing multiplicative function is exponential

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing function such that $$f(2)=2$$ and $$f(mn)=f(m)f(n)$$ for all positive integers $m,n$ such that $$\gcd(m,n)=1$$ Prove that $f(n)=n$ ...
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76 views

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime?

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime? Note, $f$ is multiplicative and $\sigma(n)>1, \;n>1$. Therefore $f(n)$ is prime only when $n=p^\alpha$, with $p$ prime, $\alpha\geq1$. ...
2
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0answers
27 views

Closed-form for $F(n) = \sum_{d|n} \omega_x(d) $

Given $x\in \mathbb{C}$ let's define the function $$\cases { \omega_x(n)=1, \quad n=1 \\ \omega_x(n)=x^r, \quad n=p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r}}$$ a) Prove that $\omega_x(n) \colon \mathbb{...
2
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1answer
39 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{...
2
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0answers
67 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 $\sigma_0$ ...
2
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62 views

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 ...
2
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1answer
108 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). ...
2
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0answers
22 views

Relation for multiplicative functions

I have some problems understanding relations between convolutions and Euler product. I want to express as a convolution or an additive formulation for: $$N\varphi(N) \prod_{p | N} (1+p^{-1})$$ ...
2
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1answer
485 views

show that $ \sum_{p \leq x} \log p \big( \log x - \log p\big) = O(x) $

Show that: $$ \sum_{p \leq x} \log p \big( \log x - \log p\big) = O(x) $$ This is not an exercise. This is implied in one line of proof by Atle Selberg. Additionally the paper asks to show: $$ \...
2
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0answers
345 views

sum of divisors function $\sum \tau(n) = \frac{1}{4}$

These notes on multiplicative number theory mention the convolution $ 1 \ast 1 = \tau$ (where $\tau$ is the divisor function not Ramanujan tau function. Therefore $$ \bigg(\sum \frac{1}{n^s} \bigg)^...
2
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1answer
30 views

Multiplying fractions with an x value

$\left(\sqrt{4+\frac{1}{x}}-2 \right) \cdot \left(\sqrt{4+\frac{1}{x}}+2\right)$ I get $\large\frac{1}{x}$ because the square roots go away and the $2$s multiply to make $-4$, so it's: $4 + \large\...
1
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2answers
73 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 ...
1
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1answer
26 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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0answers
39 views

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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73 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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0answers
44 views

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}...
1
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1answer
164 views

Let $k$ be a fixed positive integer. Prove $f(n)=gcd(n,k)$ is multiplicative.

This is what I have to far... Let $f(x)=g \implies gcd(x,k)=g$ We can then write, $ax+bk=g$ Let $f(y)=h \implies gcd(y,k)=h$ We can write, $cy+dk=h$ Multiplying, $(ax+bk)(cy+dk)=gh$ $\...
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0answers
45 views

Infinite Product for Multiplicative Functions

Do there exist closed form functional representations for the following products?: $$ f(x,\alpha) = \prod_{n=1}^\infty \left( 1 + \alpha (n) \ x^n \right) $$ Where $\alpha$ is some multiplicative ...
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47 views

Can't find the error with the following sum involving multiplicative functions

I apologize for this question which might be trivial but I'm stuck with this issue and I'm probably doing some mistake over and over again. Here's the framework. Let $f$ and $h$ be two multiplicative ...
1
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1answer
77 views

Looking applications for these statements involving multiplicative functions and Euler-Fermat theorem

It is well knwon that for positive integers such that $gcd(a,n)=1$, Euler-Fermat Theorem states than $a^{\phi(n)}\equiv 1 \mod n$, where $\phi(n)$ is the Euler totient function counting positive ...
1
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1answer
28 views

$a\cdot(b^{-1}\bmod m)$ Can be be solved using modular multiplication

Does $a\cdot(b^{-1}\bmod m) = (a\bmod m) \cdot(b^{-1}\bmod m).$ where $\bmod$ represents remainder left on division with $m$. $b^{-1} \bmod m$ is multiplicative inverse.
1
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1answer
245 views

Prove $\lambda(n)=\sum_{d^2|n}\mu(n/d)^2$ and $\mu^2(n)=\sum_{d^2|n}\mu(d)$

$\lambda(n)$= $\sum_{d^2|n}$ $\mu(n/d)^2$ and $\mu^2(n)$= $\sum_{d^2|n}$ $\mu(d)$ Having a little bit of trouble here.Can I use the fact that $\sum_{d|n}\lambda(n)$ is a characteristic function for ...
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0answers
492 views

Modular multiplicative inverse and coprime numbers needed.

I have a 64 bit algorithm that uses modular multiplicative inverse and coprime numbers, and I need to convert it to 32 bit. This math is not my area, and I cannot find an online calculator, so I hope ...
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12 views

Estimation of the convolution of two multiplicative functions

Let $f,g:\mathbb{N}\to \mathbb{C}$ be two multiplicative arithmetic functions. Assume that we know an asymptotic behavior of $f$ and $g$. Is there any general result for asymptotic behavior of the ...
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14 views

Karatsuba Multiplication Rule in dividing a Number in two parts

In Karatsuba algorithm for multiplying two numbers, we divide each number into two. For example: x= 1234 y= 2456 Then a = 12, b = 34, c = 24 , d = 56 ...
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18 views

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 ...
0
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1answer
24 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 ...
0
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1answer
63 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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116 views

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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52 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 ...
0
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1answer
33 views

Solve de equation: σ(∅(n))=2^n

Solve de equation: $$\sigma(\phi(n))=2^n$$ I got $$\sigma(n)={2^{n+1}}$$ Then, if $n=p^a$, $$\sigma(n)=\frac{{p^{a+1}}-1}{p-1}={2^{n+1}}$$ Then I got stuck... I don't know how to find this prime, I ...
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0answers
183 views

Showing a nonzero multiplicative arithmetic function satisfies $f(1) = 1$

let $f$ be multiplicative arithmetic function. if there exist a positive integer $n$ such that $f(n)$ is not equal to $0$ (so $f$ is not identically zero) prove that $f(1)=1$. I got that $gcd(1,n)=1$ ...
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1answer
119 views

Dirichlet character modulo $k$

Let $f$ be a completely multiplicative function. If there exists $k$ such that $$f(n+k) = f(n)$$ for any $n \in \mathbb{N}$, then there exists $k_0$ such that $f$ is a Dirichlet character modulo $k_0$....
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1answer
39 views

Is there any way to simplify this function?

Let $\mu(x)$ be the mobius function and $\Lambda(x)$ be the vonmangolt function. $$ f(n) = \sum_{d|n} \mu(d-1)\Lambda(d - 1)$$. Is there a way to simplify or estimate this function. $f(3) = -\log(2)$ ...
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72 views

Proving the existence of a real number in a finite measure of a translation invariant

Someone please explain this: Presume that for each $x \in \mathbb{R}$ and $A \subseteq \mathbb{R}$, that $x + A = \big\{ x + a \mid a \in A \big\}$. Here, A and x + A are Borel sets for all $x \in \...
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55 views

How do I use something related to mobius inversion to solve this problem?

The problem is given below: For two sequences of complex numbers $\{a_0, a_1, \cdots, a_n, \cdots\}$ and $\{b_0, b_1, \cdots, b_n, \cdots\}$ show that the following relations are equivalent: $$a_n = \...
0
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1answer
66 views

Number of zeros of a polynomial modulo n is a multiplicative function

Let $f$ be a polynomial with integer coeffcients. For $n\geq1$ let $N_f(n)$ denote the number of pairwise incongruent solutions of $f(x)=0$ mod n. I need help proving that $f$ is a multiplicative ...
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2answers
90 views

Proving that a summation is multiplicative

I have been give a project for number theory: For $m>0$ , let $f(m) = \sum_{r=1}^m \frac{m}{\gcd(m,r)}$ . Evaluate $f(m)$ in terms of the prime factorization of $m$. So far, I have found a formula ...
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1answer
649 views

Euler-Totient Multiplicative

http://www.oxfordmathcenter.com/drupal7/node/172 By and large, I understand this proof, however I'm struggling to understand how the Chinese remainder theorem implies that there exists some $x \in S_1$...
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0answers
51 views

Summatory Function $F(n) = 1 $ for all $n$ odd, and $F(n) = 2$ for all n even

So, I have this summatory function $$ F(n)=\sum_{d\mid n}f(n)$$ that goes $F(n) = 1$ for $2\nmid n$, and $F(n)=2$ for $2\mid n$. This summatory function is multiplicative. I need to describe the ...
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0answers
131 views

Conjecture related to the Erdős discrepancy problem

Conjecture: If $k \in \mathbb{N}$ and $S$ is an infinite set of primes, then the multiplicative $\pm$-sequence generated by $S$ contains $+^k$ as a substring infinitely often. (If $S$ is allowed to ...
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1answer
259 views

Figuring out a factor of modulo multiplication knowing other factors

So the problem is this - we have a simple equation: (A * B) % N = X All numbers are large integers. We know B, N and X, is it possible for us to figure out the last factor A without checking every ...