What makes for a really good mathematical essay?

Andreas Holmstrom

Andreas Holmstrom

— 3 min read
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I added some more information over at the Essay competition page, including some examples of really good essays and some notes on what makes an essay good. Check it out and spread the word in case you have students or friends who might be interested in submitting something! Below is a copy of the entire page, for those of you who read these things as emails.

Essay competition

New Year's Eve of 2026 will be the final deadline for the first ever PeakMath Essay Competition.

The competition is open to all PeakMath members, and one or several small prizes may be awarded to the best essay(s) on any topic broadly related to L-functions. The essay may present an original solution to an unsolved problem, a numerical investigation of some beautiful and mysterious patterns, or an informative and elegant presentation of some known piece of mathematics.

Inspired by the SoME competition hosted by 3Blue1Brown, this is an invitation for you to share some of your own work with a wider audience, and an opportunity for all of us to learn new things from each other. The essays will be posted on this site under your name. If you are a student, you might also want to think about submitting it to a journal like Involve or Columbia Journal of Undergraduate Mathematics, or submitting it to a competition like Tom Rocks Maths Essay Competition or your own country's equivalent of the Norwegian Young Researchers Contest.

What to write about?

Any L-function-related topic that you personally find interesting would be a good one for an essay submission. Take something that has caught your attention in the RH Saga videos, or something at the edge of your current knowledge that you could understand better by writing up a good exposition. If you are still looking for a topic, I wrote a post on generalized Bernoulli numbers that could be a starting point.

A really good essay would be characterized by some of the qualities listed by Tao in his own essay What is good mathematics? I'll quote from his conclusion:

As we can see from the above case study, the very best examples of good mathematics do not merely fulfil one or more of the criteria of mathematical quality listed at the beginning of this article, but are more importantly part of a greater mathematical story, which then unfurls to generate many further pieces of good mathematics of many different types. Indeed, one can view the history of entire fields of mathematics as being primarily generated by a handful of these great stories, their evolution through time, and their interaction with each other.

Below are three examples of essays I loved for different reasons, for some further inspiration!

For example, the last essay tells the story of how the unsolvable quintic can in fact be solved, if we admit among our operations not just addition, subtraction, multiplication, division and roots, but also modular forms!

What are the rules?

The entire point of an essay is to deepen human understanding of a specific theme or topic. A good essay would be one that reflects your own struggle with the material, your effort to organise the material so that it makes sense to you, and preferably some computations or investigations that you have chosen and worked out for yourself. A good essay would also be one that helps the reader understand more. When I read your essay, I an hoping for insight and connections I hadn't made before, and I will ask myself if your essay is something I would recommend to my own students for them to learn about the topic.

Given this purpose, submitting something AI-generated is just a waste of everyone's time. Regarding specific rules, I will use common sense in line with the rules for the SoME competition, which you can find here.

A reasonable length for a good essay would be something between 5 and 25 pages. Beyond that, there are no specific rules - just do your best, think of the essay as a labour of love and as a gift to anyone who might come across it and read it.

How do I submit the essay?

A special submission form will be accessible here from Dec 1 to Dec 31. You will be asked to submit a pdf file and/or a link to your own webpage where the essay is hosted, together with some information about yourself and your writing process.