Asymptotics for zeta zeros?

Question

What are the best known asymptotics for the nth zeta zero (imaginary part)? Is there anything similar to $p_n\sim n\log n$, ie where $\rho$ is in form $\sigma+it$, $t_n\sim\dots?$

Answer 1

Yes, $\gamma_n\sim2\pi n/\log(n)$. This is in Titchmarsh, for example.

For a better asymptotic, write the number of zeros to height $T$ as $$ N(T)\sim \frac{T}{2\pi}\log\left(\frac{T}{2\pi}\right)-\frac{T}{2\pi}. $$ (The error term in this asymptotic is $O(\log(T))$.) We want to invert the relationship $$ n \sim \frac{\gamma_n}{2\pi}\log\left(\frac{\gamma_n}{2\pi}\right)-\frac{\gamma_n}{2\pi}. $$ Mathematica tells me that $$ \gamma_n\sim \frac{2\pi n}{W(n/e)}, $$ where $W(z)$ is the Lambert function, which inverts $z=w\exp(w)$. ($W(z)$ is called ProductLog in Mathematica)

A better asymptotic would require a better one for $N(T)$, which depends subtly on $S(T)$, the argument of the Riemann zeta function.

Answer 2

You may check following paper:

A theory for the zeros of Riemann Zeta and other L-functions by Guilherme França, André LeClair