I have to start by saying that The Arithmancer by White Squirrel is the first fanfiction I've ever read, unless you count mainstream works like The Mists of Avalon. I wouldn't have given it a chance if it hadn't been recommended by mathemusician Vi Hart. Once I started reading, I couldn't stop. Having now finished the second book in the series (Lady Archimedes), I'm ready to lay out why I think you should read it, too.
Like most people in the world, I enjoyed the Harry Potter books by J.K. Rowling (hereafter called "HP" or "canon") and the movies based on them. But also like most people, I couldn't help noticing that the wizards are not particularly bright. They're not just ignorant of non-magical ("muggle") science and technology and current events, they're genuinely incurious and don't pay attention to anything that happens in the muggle world. It's hard not to wonder what would happen if a muggle-born wizard arrived at Hogwarts School of Witchcraft and Wizardry with even an average 11-year-old's knowledge of the 20th century, and I'm sure there have been other fanfics based on that premise.
The Arithmancer takes it a step further and explores what would happen if a really gifted muggle-born student came to Hogwarts with a knowledge of math, science, history, and pop culture beyond her 11-year-old cohort. As someone who often felt like the only one paying attention in class, I can easily sympathize with Hermione Granger's frustration in canon ("Didn't you read the homework?"), and I knew some folks in high school who were nearly as well informed as Hermione is at age eleven at the beginning of The Arithmancer, so it's a stretch but not unheard of for a kid to be that well informed. In any case, she's an avatar for us readers who are older and more knowledgeable now to imagine ourselves in her shoes encountering the magical world.
The early chapters of The Arithmancer stick pretty close to the plot of the first HP book, but being from Hermione's perspective before she really bonds with Harry and Ron, they dovetail nicely with the boys' scenes. That is, Hermione will get back from an adventure on her own and find out over mealtime conversation that Harry and Ron have had their own adventure which we know about from canon. I found this very enjoyable, since the things she learns by exploring Hogwarts Castle on her own really help it to feel more real and more magical at the same time, and help me appreciate canon more, as well as laying the groundwork for important developments later.
White Squirrel does a great job (in my opinion) of walking us through Hermione's inner struggles as she tries to balance her tremendous ambition and consequently overwhelming work load with her human limitations. A lot of time is spent describing precisely why she was crying in the restroom when the troll got loose (in book 1 of canon) and why she was so shaken up after learning of the Dementor's Kiss (in book 3 of canon). Some other reviewers have complained that this vulnerable introspection makes her look weak and fragile, but it's part of growing up, and she is growing into an unbelievably strong and resilient person.
As the story goes on it diverges more from canon. Scenes and entire plot lines develop differently. Hermione learns more about the magical world and how it works than Harry ever did. Characters who died in the HP books survive here, make significant contributions to the plot, and have satisfying character arcs. Minor characters who barely had names in canon get developed into real characters with their own arcs. In fact, only a handful of characters come to mind that are less developed in this version than in canon, and I don't miss them. The dialogue and character development feel to me very true to what Rowling set out; this feels not so much like a different world as a deeper dive into the same world. And that's what my recommendation comes down to: if you want to spend more time exploring magical Britain, this is your ticket.
How much time? Weighing in at 1.2 million words, The Arithmancer and Lady Archimedes together are longer than the 7 books of canon. If you read a chapter a day, you'll be done in a little under six months! If you don't want to do all that reading on screen -- there are rather a lot of typos in the manuscript, which may or may not bother you -- there are recordings on YouTube.
One other caveat I will offer: although the death count is significantly lower in this version than in canon, some of the deaths are more gruesome. J.K. Rowling didn't give us the names or details of most of the 54 casualties in her version of the climactic Battle of Hogwarts; White Squirrel does, and it can be a little unsettling. But Hermione and the others wrestle much more realistically with the human costs of their decisions than they ever did in the original. She even goes to therapy! What a role model.
On the whole, I think anyone who enjoyed the HP books should give this story a try. I'll look forward to discussing it with you!