Author: Wojowu

Higher reciprocity laws

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!

Direct download link to the PDF

Long well-ordered chains of functions under eventual domination

\(\newcommand{\N}{\mathbb N}\)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.

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PNT for polynomials using the zeta function

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.

The proof is based on the exposition in Davenport’s Multiplicative Number Theory and requires some understanding of complex analysis.

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