# 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 ...
1 vote
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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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### 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 ...
33 views

### 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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### 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, ...
### 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+\...