Why such an interest in studying prime gaps?

Since you ask about zeta zeros, Riemann hypothesis implies the gap is $O(\sqrt{p_n} \log p_n)$.

Larger gap will give you nontrivial zero off the critical line, disproving RH.

On the other hand, bounding the gap by $O(polylog(p_n))$ will solve the open problem for deterministically finding primes in polynomial time.

For practical purposes, some cryptographic algorithms need to find prime efficiently. Large gaps may break some implementations.

There is also a recent paper "Unexpected biases in the distribution of consecutive primes" by Robert J. Lemke Oliver and Kannan Soundararajan, which resulted in some sensational headlines. The relation to prime gaps is currently under discussion on math.SE.

A new (strong) result may affect proving Legendre's conjecture

The first thing you can read there is "prime gaps".