# Abstraction

#### Author: Wojowu

Today I wanted to share with you a write-up of a talk I have given to other PROMYS Europe 2018 counsellors about two weeks ago. The topic of the talk was higher reciprocity laws, which is a general term used to describe a great variety of results which, in one way or another, can be seen as generalizations of the classical law of quadratic reciprocity. I have tried to give a somewhat motivated account of how each and every one of those generalizes the previous ones and, ultimately, how all of them are just special cases of one theorem (or of a single, even more general conjecture!)

This is a topic I have been fascinated by for years now and I have been seeking an opportunity/excuse to read more into it. Around a year ago I have given a 30-minute talk on this topic during a student conference I was attending, but due to time constraints I couldn’t get far into the topic nor did I get deep into it myself. My talk during PROMYS Europe was different – I would like to thank all counsellors for having patience to sit through my 2.5 hour long (!) talk, despite my promises I will keep it reasonably short. I have greatly enjoyed reading up on the topic and then preparing to present it to others.

I must admit I am quite proud of the write-up for this talk, which is why I would like to share it with everyone. Except for some minor fixes, this is a version which ended up in our yearbook, but I hope some people find some benefit in this, perhaps even find this topic as interesting as I do!

$$\newcommand{\N}{\mathbb N}$$Consider the set $$\N^\N$$ of all functions from $$\N$$ to itself. We impose a partial order of eventual domination on this set: we say that $$f$$ eventually dominates $$g$$ if for large enough $$n$$ we have $$f(n)< g(n)$$. In this poset, we consider chains, in particular the well-ordered ones. One question we can ask is, how long can such chains be? Here we establish a lower bound $$\omega_2$$ for the supremum of possible lengths.

All background necessary for this post is some basic knowledge about ordinals, in particular cofinalities.

In this blog post of mine I talk about the “polynomial zeta function” $$\zeta_q(s)$$ and prove an analogue of Riemann hypothesis for it (indeed, this zeta has no zeroes at all). At the end I mention that this zeta function is, in fact, the “correct” zeta function for this purpose. In this blog post I am going to derive a formula analogous to the von Mangoldt’s explicit formula. This is a little project for me to see whether I’m able to work out all the details.