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?
[Math] How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis if at all
elementary-number-theorynumber theoryprime numbersriemann-zeta
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Too long for a comment : I believe that its origins will be forever lost in the mist of time,
for the following very simple reason :
Euler was already aware in the eighteenth century that all primes except for $2$ and $3$ are of
the form $6n\pm1.~($At any rate, such a trivial statement is relatively easy to either discover
or understand, even by people with only the most basic mathematical knowledge$).~$
Then the next question which naturally arises is about the density of those “lucky” values of
n for which both neighbors of $6n$ are simultaneously prime. So basically all that's left to do,
after first dispensing with certain formalities pertaining to what is considered to be academi–
cally acceptable mathematical etiquette, such as actually proving that their number is indeed
infinite $($most likely by using some painfully obvious argument based, say, on reduction to
the absurd, and the like : as in the case of proving that there are an infinite number of primes,
for instance$),~$ would be getting down to the really hard part of actually quantifying their
frequency, and then venturing to offer a mathematical explanation for the experimentally
obtained results $\ldots$
Except that —oh, wait a second— remember that first “easy” half we were talking about just
earlier ? Well, as “luck” would have it, it turned out to be not so easy after all $\ldots$ So that's it,
in a nutshell.
The distribution of primes is not going to change. No matter what we discover about the Riemann hypothesis or any other area of math, the distribution of primes will not change.
The Riemann hypothesis implies a bound on the error term in the prime number theorem. Specifically, it implies that $\pi(x)=\frac x{\log x}+O(\sqrt x\log x)$. If the Riemann hypothesis is shown not to be true, then we will not know that this result is true. (I believe, though I may be wrong, that the result is implied by but not equivalent to the RH; correct me if I'm wrong.)
Now, any theoretical proof that the RH is false (or true) would almost certainly involve theory which would cast further light on the distribution of the primes in some regard which might be more valuable than the disproof (or proof) of the RH itself. A discovery of a zero not on the critical line would of course be less helpful in this regard.
In either case, it is unlikely that the RH has any direct connection to the twin prime conjecture, since twin primes occur with frequency at most $\frac 1{\log x}$ in the primes (Brun's theorem), so a bound on the error term in the prime number theorem is probably too specific a result to have much effect. The twin prime conjecture is more closely related to the occurrence of primes in polynomials, and so to conjectures such as Schinzel's Hypothesis H or the Bateman-Horn conjecture (each of which imply the twin prime conjecture) or Bunyakovsky's conjecture (a weaker version of the above two which does not).
Best Answer
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!