"Sir Michael Atiyah claims to have demonstrated a simple solution to the Riemann hypothesis".
If solved, what results we will have in Cryptography?
Nothing. The Riemann hypothesis would have some implications about the distribution of prime numbers. No cryptosystem uses the distribution of prime numbers for security, so there would be no effects on any current cryptosystem.
One cryptosystem in particular, RSA, uses prime numbers. But it doesn't use the distribution of prime numbers in any way, just the difficulty of factoring composite numbers which are the product of two large primes.
As a (strained) analog it would be like thinking that finding some information about the relation of total roadway length to population in countries and wondering how it would change mapping. Sure, maps draw roads, and countries, but the statistical information isn't relevant.