The Meaning of R's non-Trivial Zeros

Is there an intuitive explanation for why the Riemann zeta Function (rather than hypothesis) contains interesting information about the distribution of primes in language comprehensible to a non-mathematician?

Related questions might be...

What does a non-trivial zero indicate in respect of the distribution of primes? What do we learn from a zero when we find one?

I'm not sure, but it might also be same question if I ask how the input to the zeta function is related the output, or how the input value relates to locations on the number line.

But I'm muddled, as you can tell, so the related questions may be a red-herring. It's mainly the first question that I'm trying to answer.

Many thanks for your patience. Imagine I'm twelve years old and not Gauss.

• Imagine $\psi(x) = \sum_{p^k \le x} \log p$ as a different version of the prime counting function $\pi(x) = \sum_{p \le x} 1$. The main point is the explicit formula $\psi(x) = x- \sum_{\rho \text{ non trivial zeros}} \frac{x^\rho}{\rho} +f(x)$ where $f(x) = \frac12\log 2\pi-\sum_{k=1}^\infty \frac{x^{-2k}}{-2k}$ is almost constant. So from the non trivial zeros we know $\psi(x)$, and from the RH we know that for any $\sigma > 1/2$ then $\frac{\psi(x)-x}{x^{\sigma}} \to 0$. Until now we only know the prime number theorem saying it is true for $\sigma \ge 1$. – reuns Feb 9 at 16:57
• @reus -= Thank for trying but I can't see how this answers the question. I did warn you that I'm not a mathematician but I expect you've forgotten long ago what this is like. Thanks anyway. – PeterJ Feb 10 at 11:34
• Because there are no other ways to answer your questions. You are supposed to look at each term $\frac{x^\rho}{\rho}$ and see how its growth rate depends only on $\Re(\rho)$. If some zeros of $\zeta(s)$ have real part $\ge \sigma$ then $\psi(x)-x$ grows at least as fast as $x^\sigma$. If the RH is true then $\psi(x)-x$ grows just a little faster than $x^{1/2}$. – reuns Feb 10 at 21:18
• Let me try to help out with the 'how this answers the question' part. We start with the accepted answer to your previous question. It says '$\pi(x) \approx \int_2^x \frac{1}{\ln(t)} dt$ and the zeroes of $\zeta$ tell you what this "\approx" means, i.e. how far the left hand side can stray from the right hand side, expressed in terms of $x$'. Now given that answer there are two separate followup questions we could ask: 1) HOW exactly appear the zeroes in the description of how far the left and right hand side can differ? Do I add them? Multiply them? Subtract them? (ct in next comment) – Vincent Feb 10 at 21:35
• @Vincent - I'm able to follow all of this and it's very helpful. I need to do some thinking before saying any more. I think I'm slowly approaching the question I should have asked in the first place. . . – PeterJ Feb 12 at 18:39