How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis if at all?

How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?

Solution 1:

Edit 2: For a more tenuous connection between these two theorems, one person whose name might crop up someday for both (please note I am just speculating here!) is that of Fields Medalist Atle Selberg. His Selberg sieve underlies some of Zhang and his contemporaries' work on the weakened version of the Twin Prime Conjecture (see Edit 1 below); meanwhile, the Selberg trace formula may (this is the speculation part) someday be used in a proof of the Riemann Hypothesis. For these latter connections, you could either look up `The Selberg trace formula and the Riemann zeta function' in google scholar or note that Paul Cohen strongly believed this might be a way to get at RH. Cohen's beliefs are alluded to in the AMS piece on his passing, e.g., by Peter Sarnak, though one must note that Cohen spent much of his post-CH life stagnating on this problem. Anyhow, as for connections between these two works of Selberg: None is apparent to me other than their originator; but perhaps this is fodder for another MSE question.

Edit 1: Note that the most recent progress on a weakened version of the Twin Prime Conjecture made use of the Riemann Hypothesis for varieties over finite fields (see Yitang Zhang's pre-print, available here, p. 6). That is, it made use of the Weil Conjectures; in particular, using methods from Deligne's proof rather than Dwork's (low-level: see here; high-level: see here). For more on YTZ's recent work, see the evolving question/answers on MO here.

A couple quotations from Dan Goldston in this paper:

"While the Riemann Hypothesis is decisive in determining the distribution of primes, it seems to be of little help with regard to twin primes."

"The conjecture that the distribution of twin primes satisfies a Riemann Hypothesis type error term is well supported empirically, but I think this might be a problem that survives the current millennium."

However, the first part of your question is more general, in that you ask about how proving one of these conjectures/hypotheses would affect the other. That is a more nebulous question, because it's hard to predict what sort of machinery will ultimately be developed to resolve these questions.

If this answer sounds like a bit of a downer to you, perhaps there is a silver lining: given two large conjectures that don't really have much bearing on one another, it would/will be interesting to see how the resolution of both could be applied elsewhere. If they were intimately connected, solving one might knock the other one off; with their relation tenuous at best, it ought to take some wonderful mathematics to dispose of both. It remains to be seen what can be done with the machinery resulting from each - I just hope it is seen in our lifetimes!

Solution 2:

Not too much, I don't belive that Twin PRime suggest much for Riemann (unless its some statistical stuff or basic guarantees).

The Twin Prime Conjecture is really part of a series of other conjectures which looks follows.

1. Twin Prime: Are there infinite primes with difference 2?
2. De Polignac: Are there infinite primes with an even difference?

3. Are there infinite consecutive primes with an even difference? aka CAN EVERY even integer be expressed as the difference between 2 consecutive primes (this is a small step towards solving the GoldBach Conjecture which itself has some statistical effect on Riemann)?

4. From there the next step is to then determine if every possible arithmetic progression is included in the list of primes (Excluding those that violate certain basic rules ex: 2 2 2, which I belive has been proven)

5. Then is every valid consecutive arithmetic progression possible (ex: does 2 6 6 6 6 occur an infinite number of times for consecutive prime differences? Nope b/c 2 mod 5 + 6 mod 5 *3 = 0 mod 5 ---> this isn't possible but other subsequences such as 2 4 may be possible instead!)