The Riemann Hypothesis [duplicate]

(EDIT: I've marked this question as answered in order that I can go away and come up with a better one. Thanks to everybody for the helpful answers.)

Is it possible to describe the RH in language comprehensible to a non-mathematician?

In my experience (as a non-mathematician) the answer is a definite 'no'. But before I give up on it I thought I'd ask for some views.

By 'explain' I do not mean explain fully. I mean explain the basic structure of the hypothesis. Something like - we feed values into a function and the output is plotted to create a landscape and a critical line appears and we conjecture that all zero outputs fall on this line. I expect even this is wrong but it's this level of explanation that I'm after, with just a bit more detail added.

I've spent a long time trying to understand the relationship between the inputs and outputs of the Zeta calculation and got nowhere. Yet I'm not stupid. There's something I'm missing but none of the explanations I've been offered allow me to understand what it is.

I know a bit about the primes but must ask for answers in a more or less natural language or will not be able to cope. I know such requests annoy mathematicians but I can only ask.

I have read Du Sautoy, Derbyshire and much more but never found an explanation that doesn't assume I'm a graduate mathematician. I need a children's book on the topic that describes it in very general terms.

Is such a thing possible?

EDIT: Vincent has suggested a similar but perhaps interestingly different question. This is - is there an intuitive explanation why the Riemann zeta FUNCTION (rather than hypothesis) contains interesting information about the distribution of primes in language comprehensible to a non-mathematician?

My intuition is that the explanation would be the harmonic series and its role in determining the distribution of primes. I'll hesitatingly ask if this makes any sense. Or are there other reasons? As it happens I'm coming at this as a musician and have an interest in the musical aspect of the distribution or primes, or rather in the distribution of non-primes that determines the primes.

EDIT 2: I see that similar questions have been asked previously. Thanks to the members who have provided links to them. I've been checking them and so far they have been helpful but not so much as to persuade me to take this one down. This may change as I keep reading and I'll withdraw this one if so.

marked as duplicate by Did, davidlowryduda♦Feb 8 at 13:03

• Even though I am a mathematician I VERY MUCH empathize with the poser of this question. I think the best way too understand the importance of/interest in/usefulness of the RH is through the connection to the prime number counting function as explained by AlgebraicsAnonymous below and more elaborately in the accepted anwer to the question pregunton links to above. But then of course there is a follow up question: is there an accessible way to see that the the RH relates to prime numbers in the first place? (ctd in next comment) – Vincent Feb 8 at 13:03
• (Apparently pregunton removed their comment with the link to the other question. I have now put the link below.) What I wanted to say: once we are at that point we might ask a simpler (but still very hard) question: is there an intuitive explanation why the Riemann zeta FUNCTION (rather than hypothesis) contains interesting information about the distribution of primes in language comprehensible to a non-mathematician? I would be really interested in an answer to that since even if I am mathematician enough to follow the proof, intuition fails me here. – Vincent Feb 8 at 13:07
• Here is the link to the other question again: math.stackexchange.com/questions/7981/…. – Vincent Feb 8 at 13:10
• @Vincent - Thanks. To be honest even some sympathy is appreciated. I've already learnt a little from the answers so progress is being made. I've followed your link and found it helpful, but not so much I want to close this one. I like you secondary question about the function and may edit my question to include it. I assume the connection is the harmonic series but this is grasping at straws. – PeterJ Feb 8 at 13:19
• Clicking some more 'linked question'-links I found the following question and answer that come quite close to what I had in mind: math.stackexchange.com/questions/2144940/…. It sort of answers the question 'what does the Riemann-zeta function at $s = 1$ (so no need to write the $s$ at all) have to do with the distribution of prime numbers, from which it is a very small step to my question of 'what does the Riemann-zeta function with the distribution of prime numbers?' – Vincent Feb 8 at 16:26

The prime numbers have density approximately $$1/\ln(n)$$, in the sense that the probability that a number $$n$$ is prime is approximately $$1/\ln(n)$$. From this, one would guess that if $$\pi(x)$$ (for a number $$x> 0$$) denotes the number of primes below $$x$$, we would have $$\pi(x) \approx \int_2^x \frac{1}{\ln(t)} dt.$$ The Riemann hypothesis asserts that the error in this approximate equation does not exceed $$\frac{\pi}{8} \sqrt{x} \ln(x)$$.

In short: The Riemann hypothesis asserts that the prime numbers are very strictly distributed according to the density $$1/\ln(n)$$.

• Thank you. So where there is a non-trivial zero does this represent a distribution or output that is bang in line with the PMT? – PeterJ Feb 8 at 12:46
• @PeterJ Did you ever see explicit formulas $\psi(x) = x-\sum_\rho \frac{x^\rho}{\rho} - C-\sum_{k \ge 1} \frac{x^{-2k}}{-2k}$ where $\psi(x)$ is just a different version of $\pi(x)$. That sum over infinitely many non-trivial zeros can be truncated as $\psi(x) = x-\sum_{|Im(\rho)|< T} \frac{x^\rho}{\rho}+C+O(x^{-2}+\frac{x^{\sigma_0}\log^2 x}{T})$, there are about $AT\log T$ terms in the sum and $\sigma_0 =\sup\Re(\rho)$. It's technical but from this it is perfectly clear how $\Re(\rho)$ determinates the growth of $|\psi(x)-x|$, that's how Riemann's 1859 paper revolutioned the primes – reuns Feb 8 at 12:53
• @reuns - Many thanks, but this over my head. – PeterJ Feb 8 at 15:59

It's tempting to try to satisfy you with a "simple" claim known to be equivalent to the Riemann hypothesis. To my mind, the best example is this: for an integer $$n\ge 5041$$, the sum of the positive factors of $$n$$ is less than $$ne^\gamma\ln\ln n$$. As long as I can teach you what natural exponentials and logarithms are (which isn't even undergrad material), and explain the Euler-Mascheroni constant $$\gamma:=\lim_{n\to\infty}(-\ln n+\sum_{k=1}^n\frac{1}{k})$$, I've technically given you a full understanding of what it means for the hypothesis to be true or false.

The problem with this approach, however, is it doesn't make clear why anyone cares if it's true or false. For that, we need to discuss the consequences of the hypothesis, and I doubt I can prove any interesting ones to you starting from the formulation I just gave, without some heavy-handed material (you know, the kind that proves equivalence with a "non-lay" formulation of the hypothesis).

Nor will I have made clear why anyone thinks it's probably true. After all, its consequences are irrelevant if it's false.

Roughly speaking, the story is this:

• There is a unique way to "analytically continue" the usual $$\sum_{n\ge 1}n^{-s}$$ definition for $$\Re s>1$$ to cover any $$s\in\Bbb C\backslash\{1\}$$. It's unique, because that's a fact about analytic continuation. The hypothesis says this continuation's roots are either negative even integers ("trivial") or solutions of $$\Re s=\frac12$$ (this locus is the critical line).
• The hypothesis is likely to be true partly because so many roots have been identified and they all fit, partly because we can prove lesser results such as there being zero density of nontrivial roots outside the critical line, partly because equivalent claims also have no known counterexamples despite heavy searching, partly because the hypothesis passes a lot of sanity checks, and so on.
• It matters, because, if it's true, we learn a lot more about how prime numbers are distributed and how quickly some important functions grow, among other things.

I don't know, however, how to "dumb it down" to the point where you don't have to learn a lot of new maths to follow the details. Nor would I be surprised if this is unfeasible. After all, there's only so far primary-school maths can get you, and ditto with secondary-school maths, and so on.

• Maybe add the fact that the Euler-Mascheroni constant is just some specific number approximately equal to $\gamma=0.577..$. – quarague Feb 8 at 13:00
• Thanks. Still over my head but I do grasp why RH is important and why we should care. It's an odd situation. I've been trying to understand what can be understood by a layman and what actually requires complex mathematics, but it seems difficult to make this distinction. . – PeterJ Feb 8 at 13:32
• @PeterJ Yes, it's a tricky one. A word of caution: talk in this context about how complicated the mathematics may get, not how complex. Apart from how a grammarian would distinguish these, to a mathematician "complex" refers to the role of complex numbers in this problem. Of course, the hypothesis is fundamentally about complex numbers, even if one can present real-only ("elementary") equivalents such as the inequality I discussed. – J.G. Feb 8 at 13:40
• @JG - Yes, I see that the mistake. It's difficult for a layman not to appear to be a complete idiot in such discussions. It's difficult even to find the right questions. – PeterJ Feb 8 at 15:57
• @PeterJ Oh, don't feel so bad. The history of mathematics is one in which even great minds often didn't or couldn't ask the right questions until surprisingly recently. – J.G. Feb 8 at 16:14