# Accessible explanation of the link between the distribution of primes and the Riemann zeta function.

I am looking for an accessible explanation of the link between the distribution of the primes and the Riemann zeta function.

I have read the related questions and answers here (eg this), and also via internet search, and also key recommended books (popular maths, like Prime Obsession, texts like Apostol, etc) - and they either gloss over the key points, or are aimed at readers with university level maths.

I would appreciate answers here, or pointers to explanations elsewhere.

My students understand the Euler product formula, complex functions, calculus.

Note - this is a repeat of a previous question (now deleted) as it was marked negative without explanation.

• I gave a negative explanation in a comment, did you read it, what was your answer? Mar 24, 2021 at 3:39
• Again, what don't you like in my answer? Mar 26, 2021 at 18:48

$$\frac{-\zeta'(s)}{s\zeta(s)}$$ is the Laplace transform of $$\psi(e^u)$$.
If $$\psi(e^u)-e^u=O(e^{\sigma u})$$ then $$\frac{-\zeta'(s)}{s\zeta(s)}-\frac1{s-1}$$ is analytic for $$\Re(s)>\sigma$$. The converse is the purpose of the Tauberian theorems, highly non-trivial, the topic of several textbooks.
• the PNT $$\psi(e^u)-e^u=o(e^{u})$$ (following from that $$\frac{-\zeta'(s)}{s\zeta(s)}-\frac1{s-1}$$ is analytic on $$\Re(s)\ge 1$$ and shifting the inverse Laplace transform integral to the left),
• that under the RH $$\psi(e^u)-e^u=O(e^{u/2} u^2)$$ (shifting the inverse Laplace transform integral even more to the left after proving a lot of non-trivial facts on $$\zeta(s)$$)