# On an extension of the Landau-Gonek formula

## TL;DR

This paper extends the Landau-Gonek formula, enabling unconditional results on the zeros of the zeta function, including simplicity and pair correlation estimates, without relying on the Riemann hypothesis.

## Contribution

It provides an extension of the Landau-Gonek formula and derives zero distribution results unconditionally, weakening the dependence on the Riemann hypothesis.

## Key findings

- At least two-thirds of zeta zeros are simple under a zero density hypothesis.
- Unconditional pair correlation estimates for zeros are established.
- Results are independent of the Riemann hypothesis.

## Abstract

We prove an extension of the Landau-Gonek formula. As an application we recover unconditionally some of the consequences of a pair correlation estimate that previously was known under the Riemann hypothesis. As one corollary we prove that at least two-thirds of the zeros of the zeta function are simple under a zero density hypothesis, which is weaker than the Riemann hypothesis. The results in this paper can be viewed as pair correlation estimates independent of the Riemann hypothesis.