Asymptotic formula of the number of Newton polygons

TL;DR

This paper derives an asymptotic formula for counting Newton polygons, revealing oscillatory behavior linked to the zeros of the Riemann zeta function, thus connecting combinatorics with deep number theory.

Contribution

It provides the first asymptotic enumeration of Newton polygons and uncovers oscillatory terms related to the Riemann zeta function zeros.

Findings

Asymptotic formula includes oscillatory terms from zeta zeros

Enumeration is based on a recurrence relation

Oscillations' amplitude estimation relates to the Riemann hypothesis

Abstract

In this paper, we enumerate Newton polygons asymptotically. The number of Newton polygons is computable by a simple recurrence equation, but unexpectedly the asymptotic formula of its logarithm contains growing oscillatory terms. As the terms come from non-trivial zeros of the Riemann zeta function, an estimation of the amplitude of the oscillating part is equivalent to the Riemann hypothesis.