On Li coefficients

Speaker: Dr Neea Palojarvi 
Affiliation: UNSW Canberra

Abstract

In analytic number theory, one of the central questions concerns on the zeros of the Riemann zeta function. For $\Re(s)>1$, the Riemann zeta function is defined as 

$$

\sum_{n \geq 1} n^{-s},

$$

and can be extended to the whole complex plane with having a pole at $s=1$. We know that the Riemann zeta function does not have any zeros for $\Re(s)\geq 1$, and the zeros with non-positive real parts are well-known. The question remains, how are the zeros with $0<\Re(s)<1$ located. This problem has many connections to other problems in number theory such as estimating the number of primes up to some height. A well-known conjecture, the Riemann hypothesis, states that all zeros of the Riemann zeta function with real part in $(0,1)$ lie on the line $\Re(s)=1/2$.

There are several equivalent conditions for the Riemann hypothesis. One of them was proved by X.-J. Li in 1997. He showed that the Riemann hypothesis holds if and only if the real numbers

$$

\lambda_n:=\sum_{\rho} \left(1-\left(1-\frac{1}{\rho}\right)^n\right),

$$

where the sum runs over the non-trivial zeros of the Riemann zeta function, are non-negative for all positive integers $n$. The terms $\lambda_n$ are called the Li coefficients. The condition has since been generalised for different functions.

The non-negativity of all Li coefficients seems to be difficult to prove. A natural follow-up question is to ask what do finitely many Li coefficients tell about the location of zeros. In this talk, I will discuss about results answering that question as well as about the Li coefficients in general. The talk is based on joint work with Samprit Ghosh and Nicol Leong.