Some Hints on Mathematical Style

Many years ago, just after my degree, I had the good fortune to be given some hints on mathematical writing by J.-P. Serre. Through the years I have found myself trying to repeat this very sound advice to other mathematicians who are also starting out. I have also been involved in the publishing of a proceedings volume, as well as being an editor of the Journal of Number Theory. Many of the papers coming my way are from young authors; so I have written down these hints in order to speed the process along. This is a slightly enlarged edition of the original “hints” of the early 1990’s. I have incorporated some recent (Spring 1998) suggestions and comments from E.G. Dunne of the American Mathematical Society and P. Vojta of U.C. Berkeley. Also (February, 2007) Serre felt that the suggestions originally presented here on what sort of computations to include in a manuscript should be changed; the current version includes these changes. These hints are presented as a source of ideas on mathematical style. Feel free to use them in any way that you deem useful.