This is a mathematical proof problem focused on divisors and number-theoretic functions. The problem requires proving several identities for a positive integer $n$, where various divisor-related functions are involved:

1. Notation used:
   • $\tau(n)$: Number of divisors of $n$.
   • $\phi(d)$: Euler's totient function of $d$.
   • $\omega(n)$: The number of distinct prime factors of $n$.
   • $d \mid n$: $d$ is a divisor of $n$.

Let’s address the proofs one by one:

(a) $\left( \sum_{d \mid n} \tau(n) \right)^2 = \sum_{d \mid n} \tau(d)^3$

• Proof Outline: This can be proved by evaluating $\tau(n)$ using its divisor definition, and explicitly expanding both sides of the equation. Recognize symmetries and relationships between divisors.

(b) $\tau(n^2) = \sum_{d \mid n} 2^{\omega(d)}$

• Proof Outline: The number of divisors of $n^2$, $\tau(n^2)$, is calculated using the factorization of $n$ and properties of exponents. Meanwhile, $\sum_{d \mid n} 2^{\omega(d)}$ represents contributions from prime divisors of $n$. The equality stems from the structure of divisor sums and how powers interact with prime factorizations.

(c) $\sum_{d \mid n} \frac{n}{d} \tau(d) = \sum_{d \mid n} d \tau(d)$

• Proof Outline: This identity reflects divisor properties and symmetry. Break the sum into parts and use $d \mid n$ to connect the left and right sides.

(d) $\sum_{d \mid n} \phi(d) = n$

• Proof Outline: This is a well-known divisor sum identity involving Euler's totient function. It states that the sum of $\phi(d)$ over all divisors $d$ of $n$ equals $n$. It can be proved using the properties of $\phi$ and its relationship to integers.

Would you like me to delve deeper into the detailed steps for each proof? Let me know which ones you'd like to start with!

Further Exploration Questions:

1. What is the general formula for $\tau(n)$, and how is it derived from the prime factorization of $n$?
2. Can you prove the relationship $\phi(n) = n \prod_{p \mid n}(1 - 1/p)$, and how does it connect to divisors?
3. How does $\omega(n)$ influence the calculation of $2^{\omega(d)}$ for divisors of $n$?
4. Why is the sum $\sum_{d \mid n} \phi(d) = n$ central to number theory and its applications?
5. Can you write a general algorithm for computing these identities programmatically?

Tip: Symmetry and properties of divisors often simplify number-theoretic proofs. Always look for connections between sums over divisors and their complements!