# Would the Riemann Hypothesis being false affect how frequently primes occur in the number system?

I want to know that if Riemann hypothesis is false (big assumption) would that lead to any effect in how frequently primes occur . Well I got this half cooked information from here: http://chat.stackexchange.com/transcript/message/21561457#21561457

And I want to get a bit more information on what was said over there.

By the way primes occur I mean that will the primes start occurring more frequently. If they start occurring more frequently, can we find out by how much? Then we might be able to relate this to the new bound which has been found between primes. That might give us a connection between the twin prime conjecture and the Riemann hypothesis.

• "No, the distribution of primes will not change, but our perception of it might." – abiessu Jun 2 '15 at 17:13
• The distribution of primes is what it is. Whether the hypothesis is true or not doesn't change how the primes are distributed. It may change "our understanding" of how they are distributed. – TravisJ Jun 2 '15 at 17:13
• So primes will start occurring more frequently?? @TravisJ – user210387 Jun 2 '15 at 17:14
• To discourage the "this is trivially false" answers/comments, you may want to clarify that you mean "how frequently we expect primes to occur" – pjs36 Jun 2 '15 at 17:23
• When we discovered that the earth was spherical (ellipsoidal), did anything happen to the earth? No, only the maps changed, and some heads were cut off (luckily people are more peaceful about $\mathsf{RH}$). – GPerez Jun 2 '15 at 17:49

## 2 Answers

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).

No of course not. Distribution of primes is set in stone. If the Riemann Hypothesis being false effects the distribution of prime numbers that implies the Riemann hypothesis is true.

• But in that chat session I was told that primes will start occurring more frequently Is that right?? – user210387 Jun 2 '15 at 17:19
• It will give us a better understanding of the prime distribution that's all – alkabary Jun 2 '15 at 17:20
• So what was told in that chat session was wrong ?@alkabary – user210387 Jun 2 '15 at 17:21
• This answer basically says that the distribution of prime numbers (which is a thing that exists and is well-defined, independently of our knowledge of it) will not change: it is what it is. What may change is the realization that our knowledge of/assumptions on this distribution could be wrong, i.e. what we thought this distribution was turns out not to be what it actually is. In short: whether the RH is true will not change the distribution, it may change what we write about this distribution in textbooks and on Wikipedia. – Clement C. Jun 2 '15 at 17:32
• Read the above: the primes will not change. The formulas we use to say how frequently they appear may change. (That is, we may realize than they happens more frequently (for some range of the reals) than what we previously thought they would) – Clement C. Jun 2 '15 at 17:37