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

Question

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.

Answer 1

$\frac{-\zeta'(s)}{s\zeta(s)}$ 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 main results are

• 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 $\zeta(s)$)

• and the Riemann explicit formula.