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Compute Edin (1)(d).

Compute Edin (1)(d).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Educational Content: Mathematical Expression Analysis #### Problem Statement: Compute \( \sum_{d \mid n} \sigma \left( \frac{n}{d} \right) \varphi(d) \). #### Explanation: This problem involves understanding and computing a sum over the divisors of \( n \). Let's break down the components of the given expression: 1. **Summation Notation \( \sum_{d \mid n} \)**: - This notation indicates that you sum the expression following it for every divisor \( d \) of \( n \). 2. **\( \sigma \left( \frac{n}{d} \right) \)**: - The function \( \sigma(x) \) (sigma function) typically returns the sum of the positive divisors of \( x \). 3. **\( \varphi(d) \)**: - The function \( \varphi(x) \) (Euler's totient function) counts the number of positive integers up to \( x \) that are relatively prime to \( x \). #### Detailed Steps: - **Step 1:** Identify all divisors of \( n \). - **Step 2:** For each divisor \( d \) of \( n \), compute \( \frac{n}{d} \). - **Step 3:** Apply the sigma function \( \sigma \) to the result of \( \frac{n}{d} \). - **Step 4:** Apply the Euler's totient function \( \varphi \) to \( d \). - **Step 5:** Multiply the results from steps 3 and 4 for each divisor \( d \). - **Step 6:** Sum all these products to get the final result. This combinatorial and number-theoretic approach requires familiarity with divisor functions and the properties of the sigma and totient functions. It's an interesting problem that showcases deep connections in number theory.
Transcribed Image Text:### Educational Content: Mathematical Expression Analysis #### Problem Statement: Compute \( \sum_{d \mid n} \sigma \left( \frac{n}{d} \right) \varphi(d) \). #### Explanation: This problem involves understanding and computing a sum over the divisors of \( n \). Let's break down the components of the given expression: 1. **Summation Notation \( \sum_{d \mid n} \)**: - This notation indicates that you sum the expression following it for every divisor \( d \) of \( n \). 2. **\( \sigma \left( \frac{n}{d} \right) \)**: - The function \( \sigma(x) \) (sigma function) typically returns the sum of the positive divisors of \( x \). 3. **\( \varphi(d) \)**: - The function \( \varphi(x) \) (Euler's totient function) counts the number of positive integers up to \( x \) that are relatively prime to \( x \). #### Detailed Steps: - **Step 1:** Identify all divisors of \( n \). - **Step 2:** For each divisor \( d \) of \( n \), compute \( \frac{n}{d} \). - **Step 3:** Apply the sigma function \( \sigma \) to the result of \( \frac{n}{d} \). - **Step 4:** Apply the Euler's totient function \( \varphi \) to \( d \). - **Step 5:** Multiply the results from steps 3 and 4 for each divisor \( d \). - **Step 6:** Sum all these products to get the final result. This combinatorial and number-theoretic approach requires familiarity with divisor functions and the properties of the sigma and totient functions. It's an interesting problem that showcases deep connections in number theory.
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