I can't point exactly to any particular thing, but I can tell you how to identify them! The magic words are "Assuming the Riemann Hypothesis..."

Any time you see somebody say "We assume RH," or "assuming RH," then any stepping stone to RH makes this assumption more reasonable/likely/anticipated. At this point many working mathematicians will hold opinions like "RH is true" or "RH is true or there's a Siegel zero" so this is kind of like assuming plate tectonics in a seismology paper, or assuming human interference in the atmosphere in a climatology paper.

In cryptography, the only times I can recall having seen it, they meant only "assuming the primes are remarkably well-behaved in their distribution." The primes are empirically remarkably well-behaved, including in the neighborhoods of typical RSA keys. So this is not surprising or unexpected, and improvements on RH should only reinforce our confidence in our empirical techniques.

Nothing because it doesn’t show a method about the distribution of prime numbers but it says that given the Riemann hypothesis is true then the leinhoff problem is true .

Pretty much all of the relationship to practical stuff in papers like this related to the RH are "it gives us information about the prime numbers, and those are used in cryptography".

It's really just about getting people to perk their ears up rather than true implications about crypto.

I think it's because any information we gain about the Riemann Hypothesis (Lindelöf hypothesis is implied by RH) gives us information about the distribution of prime numbers. Any time you gain information about the distribution of prime numbers you immediately gain information that can be applied to any form of cryptography that makes use of prime numbers. You could use this information either to break existing forms of cryptography faster, or apply it to building newer and stronger cryptography.