There are three different levels of reality in this novel: On the one hand it is the story of Terje Huuse, a Norwegian mathematician undergoing a midlife crisis. That part of the story is presented through diary entries that include an extramarital affair he has along with his attempts to write a book about the famous mathematician Bernhard Riemann. That is framed at a higher level by the story of his family who has found his diary after his disappearance. And, within all of that, there is the story of Riemann himself, focusing more on his personal life than on his mathematical research. The three levels are tied together in an interesting way by the titular imaginary number i. Riemann's research into the distribution of primes involves complex numbers, there is a sense in which the woman with whom Huuse has an affair is like this number in that they should not exist and yet they do, and finally the family is skeptical of what is written in the diary thinking that it may well be the product of Huuse's imagination. Another connection is that both Huuse and Riemann suffered from depression. Norwegian Fields Medal winner Atle Selberg also makes an appearance in the novel.
I learned about this book from Peter Hertz who told me about the German translation by Gönther Frauenlob that was published in 2007. (Danke sehr, Peter!) The title for the German version was changed to Die Riemannsche Vermutung ("The Riemann Hypothesis"), presumably to emphasize the connection to the famous German mathematician.
Disclaimer: I have not actually read this book (and since I am not fluent in either Norwegian or German, I wouldn't get much out of it if I did). So, all of the information above was put together from the bits I could learn about it from a thorough internet search along with the information provided by Peter Hertz. I apologize if any of it is mistaken or misleading. If you have actually read this work of mathematical fiction, please write to let me know!
Contributed by
Tom Louis Lindstrøm
As I have actually read the original Norwegian version of the book, I thought I should contribute a little. The description above is very accurate, but I would like to add that the novel's ending creates yet another layer of mystery. It is extremely open with a lot of possible interpretations (carefully prepared by the author), but all the ones I have found seem implausible in one way or another – it as if one is banging one's head against yet another instance of something that seems impossible, but still has to exist (as I suppose the square root of minus one must have looked to mathematicians of the sixteenth and seventeenth century). I am not against open endings, must I must admit I find this one rather irritating, but perhaps only in the way that I find a puzzle irritating until I have solved it.
The author is not a professional mathematician, but his treatment of mathematics and mathematicians seems almost flawless to me. As most of the text is written in the form of a diary by a professional mathematician, there are lots of things to get right from mathematical concepts to mathematics courses and anecdotes about mathematicians, and it all seems spot on to me. Even the description of the Math Department at the University of Oslo is impressively accurate (although it is not true that we once had blackboards framed on the inside of our office doors  that was in the neighboring building). It may seem a little strange that the protagonist has to start from scratch in his work on Riemann's biography (e.g., he seems unaware of the scientific biography by Laugwitz although he could have found it in his department's library), but that is within the bounds of poetic license.
Atle Næss is an experienced author who in addition to 14 novels on contemporary and historical themes has written biographies of Galileo Galilei, Edvard Munch, Leonardo da Vinci, and Martin Luther plus a number of books on other topics. One of my favorites is "Et hav av språk" (An Ocean of Languages) which is written in collaboration with his linguist daughter Åshild Næss, and which describes linguistic research in terms very similar to mathematical research (except that mathematicians seldom have to deal with the practical problems of travelling to and surviving on an isolated Pacific island). Næss's experience as a biographer must have helped him a lot in the work on "Roten av minus en", not only in the descriptions of the perplexities of a novice biographer, but also in getting all the mathematical details right.

