LECTURE NOTES 0 FOR CAMBRIDGE PART III COURSE ON “ELEMENTARY METHODS IN ANALYTIC NUMBER THEORY”, LENT 2015
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These are rough notes explaining some preliminary details, mostly practical arrangements, basic notation, the course synopsis, and a few background facts, for the “Elementary Methods in Analytic Number Theory” course. 1. Practical matters Lecture notes. As discussed later, this course will be divided into three main chapters. I will produce notes for each block of lectures, and post them on my webpage https://www.dpmms.cam.ac.uk/~ajh228/ at some point, probably around the end of that block of lectures. I will try to write the notes carefully, but their main purpose is for my own reference when giving the lectures. So please come to the lectures, and take your own notes whilst there! Books. The book that corresponds most closely to this course is Friedlander and Iwaniec, Opera de Cribro. This will closely match about the first third of the course, and say quite a lot about the remaining parts. Davenport, Multiplicative Number Theory has a good treatment of most of the middle third of the course. Montgomery and Vaughan, Multiplicative Number Theory contains some material on the first and last thirds of the course. You should be able to follow the course without access to these books, but they are certainly well worth a look if possible. The books by Davenport, and Montgomery and Vaughan, should be quite inexpensive and give a nice general introduction to analytic number theory. Example sheets. I expect to write three examples sheets for the course, and probably have three examples classes this term followed by a revision class in Easter term. I will post the examples sheets on my webpage https://www.dpmms.cam.ac.uk/~ajh228/ as I write them, and during the lectures we will agree a time for the examples classes. 2. Notation and conventions This is an analysis course, and will involve estimating/bounding various quantities that are too complicated to understand exactly (or, sometimes, that we don’t need to understand precisely). To facilitate this we will need a bit of notation. Date: 14th January 2015. 1
[1] J. Maynard. Small gaps between primes , 2013, 1311.4600.
[2] H. Davenport. Multiplicative Number Theory , 1967 .
[3] Hugh L. Montgomery,et al. Multiplicative Number Theory I: Classical Theory , 2006 .
[4] H. Iwaniec,et al. Analytic Number Theory , 2004 .
[5] John B. Friedlander,et al. Opera De Cribro , 2010 .