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Chapter 2: Problem 48

Show that if \(f\) and \(g\) are multiplicative, then so is \(f \star g\).

Short Answer

Expert verified
Question: Prove that the convolution of two multiplicative functions is also a multiplicative function. Solution: To prove that the convolution of two multiplicative functions \(f\) and \(g\) is also multiplicative, we showed that for any two coprime positive integers \(m\) and \(n\), the equation \((f \star g)(mn) = (f \star g)(m)(f \star g)(n)\) holds. By computing the convolution of \(f\) and \(g\) at \(mn\) and the product of their convolutions at \(m\) and \(n\), we were able to show that their expressions are equal. Thus, we demonstrated that the convolution \((f \star g)\) is also a multiplicative function.

Step by step solution

01

Write down the given information

We are given that \(f\) and \(g\) are multiplicative functions. Therefore, for any two coprime positive integers \(m\) and \(n\), we have: 1. \(f(mn) = f(m)f(n)\) 2. \(g(mn) = g(m)g(n)\)
02

Compute \((f \star g)(mn)\)

To compute \((f \star g)(mn)\), we need to evaluate the convolution of \(f\) and \(g\) at \(mn\). According to the definition of the convolution of two functions, we have: \((f \star g)(mn) = \sum_{d|mn} f(d)g\left(\frac{mn}{d}\right)\)
03

Compute \((f \star g)(m)\) and \((f \star g)(n)\)

In order to compute \((f \star g)(m)\) and \((f \star g)(n)\), we will use the definition of the convolution of two functions: \((f \star g)(m) = \sum_{d|m} f(d)g\left(\frac{m}{d}\right)\) and \((f \star g)(n) = \sum_{d|n} f(d)g\left(\frac{n}{d}\right)\)
04

Compute \((f \star g)(m)(f \star g)(n)\)

Now, we need to compute the product of \((f \star g)(m)\) and \((f \star g)(n)\). Since we already have their individual expressions, we can write their product as: \((f \star g)(m)(f \star g)(n) = \left(\sum_{d|m} f(d)g\left(\frac{m}{d}\right)\right)\left(\sum_{e|n} f(e)g\left(\frac{n}{e}\right)\right)\)
05

Show that \((f \star g)(mn) = (f \star g)(m)(f \star g)(n)\)

The final step is to show that the expressions found in Steps 2 and 4 are equal. We do this by combining the sums and using the fact that \(f\) and \(g\) are multiplicative: \((f \star g)(m)(f \star g)(n) = \sum_{d|m}\sum_{e|n} f(de)g\left(\frac{m}{d}\right)g\left(\frac{n}{e}\right) = \sum_{d|m}\sum_{e|n} f(d)f(e)g\left(\frac{m}{d}\right)g\left(\frac{n}{e}\right)\) Since \(m\) and \(n\) are coprime, the terms of the form \(de\) with \(d|m\) and \(e|n\) in the double sum are precisely the divisors of \(mn\). Thus, the sum becomes: \(\sum_{d|mn} f(d)g\left(\frac{mn}{d}\right) = (f \star g)(mn)\) Hence, we have shown that \((f \star g)(mn) = (f \star g)(m)(f \star g)(n)\) for coprime positive integers \(m\) and \(n\), which means that the convolution \((f \star g)\) is also a multiplicative function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convolution of Functions
The convolution of functions is a key concept in number theory, often used to combine two mathematical functions into one. When dealing with multiplicative functions, the convolution is a powerful method to create a new function that retains certain properties of the originals.
In number theory, the convolution of two functions, say \( f \) and \( g \), at a number \( n \) is expressed as:
  • \((f \star g)(n) = \sum_{d|n} f(d)g\left(\frac{n}{d}\right)\)
This means that the new function \( f \star g \) is built by summing over all divisors \( d \) of \( n \), multiplying the value of \( f \) at each divisor by the corresponding value of \( g \) at the quotient of \( n \) divided by the divisor.
This approach allows for a rich structure and helps in analyzing properties of numbers, such as factoring. When the original functions are multiplicative, so is their convolution. This characteristic is valuable in many areas, especially in solving problems involving divisors and primes.
Coprime Integers
Coprime integers, also known as relatively prime numbers, are those pairs of integers whose greatest common divisor (GCD) is 1. This property of integers is foundational in many areas of mathematics, especially number theory.
Coprime integers, say \( m \) and \( n \), by definition satisfy
  • \( \text{GCD}(m, n) = 1 \)
This relationship allows for some useful simplifications and factorizations in mathematical expressions.
For example, if we want to show that a particular mathematical function is multiplicative, proving it is sufficient for coprime integers. This simplifies many proofs, as demonstrated in the original exercise where \((f \star g)(mn) = (f \star g)(m)(f \star g)(n)\) when \(m\) and \(n\) are coprime.
Coprime integers are crucial in functions like Euler's totient function, defined as a count of numbers up to a given integer \( n \) that are coprime with \( n \). Their use is widespread in areas such as cryptography and modular arithmetic.
Number Theory
Number theory is a vast domain in mathematics focusing on the properties and relationships of numbers, particularly integers. It encompasses a variety of fascinating concepts, including the distribution of primes, divisors, congruences, and multiplicative functions.
In exercise problems involving multiplicative functions and convolutions, number theory provides the theoretical framework and tools. Here are some critical facets of number theory:
  • Primes and Divisors: Pieces of every number structure. Multiplicative functions often take advantage of the prime factorization of numbers.
  • Congruences and Modularity: Solve equations and problems in modular arithmetic, forming the basis of codes and cryptographic systems.
  • Multiplicative Functions: Capture the essence of number factorization, allowing the convolution of functions and demonstrating properties like being "multiplicative" over coprime numbers.
Number theory is not only the oldest branch of pure mathematics but also one of the most enduringly influential. It plays a pivotal role in modern computer science, encryption, and algorithms. Understanding concepts like convolution, coprime integers, and multiplicative functions through number theory allows for deep insights into the numbers' innate properties.

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Most popular questions from this chapter

Suppose \(n_{1}\) and \(n_{2}\) are positive integers, and let \(d:=\operatorname{gcd}\left(n_{1}, n_{2}\right)\). Let \(a_{1}\) and \(a_{2}\) be arbitrary integers. Show that there exists an integer \(a\) such that \(a \equiv a_{1}\left(\bmod n_{1}\right)\) and \(a \equiv a_{2}\left(\bmod n_{2}\right)\) if and only if \(a_{1} \equiv a_{2}(\bmod d)\).

Let \(a, b, n \in \mathbb{Z}\) with \(n>0\) and \(a \equiv b(\bmod n)\). Show that \(\operatorname{gcd}(a, n)=\operatorname{gcd}(b, n)\).

Let \(\left\\{n_{i}\right\\}_{i=1}^{k}\) be a pairwise relatively prime family of positive integers. Let \(a_{1}, \ldots, a_{k}\) and \(b_{1}, \ldots, b_{k}\) be integers, and set \(d_{i}:=\operatorname{gcd}\left(a_{i}, n_{i}\right)\) for \(i=1, \ldots, k .\) Show that there exists an integer \(z\) such that \(a_{i} z \equiv b_{i}\left(\bmod n_{i}\right)\) for \(i=1, \ldots, k\) if and only if \(d_{i} \mid b_{i}\) for \(i=1, \ldots, k\).

Let \(\tau(n)\) be the number of positive divisors of \(n .\) Show that: (a) \(\tau\) is a multiplicative function; (b) \(\tau(n)=\prod_{i=1}^{r}\left(e_{i}+1\right),\) where \(n=p_{1}^{e_{1}} \cdots p_{r}^{e_{r}}\) is the prime factorization of \(n\); (c) \(\sum_{d \mid n} \mu(d) \tau(n / d)=1\) (d) \(\sum_{d \mid n} \mu(d) \tau(d)=(-1)^{r},\) where \(n=p_{1}^{e_{1}} \cdots p_{r}^{e_{r}}\) is the prime factorization of \(n\).

Show that if both \(u\) and \(v\) are the sum of two squares of integers, then so is their product \(u v\).
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