Announcements:

Please contact the instructor if you want access to recordings of the lectures from this course.

The following lecture notes have been completed by students registered in the class:

A LaTeX template has been posted for you to use (if you wish) to start your writing assignment; it can also serve as a primer to LaTeX if you are less familiar with it. Making the .tex file compile is a first test of using LaTeX on your computer; the corresponding .pdf file has been posted as well just in case.
Lectures:
Mondays, Wednesdays, and Fridays, 10:0010:50 AM, room WMAX 216 (PIMS)
Office hours: by appointment (in person or by Skype)
Office: MATH 212 (Mathematics Building)
Email address:
Phone number: (604) 8224371
Course description:
This is a topics course in number theory, called “Analytic number theory II” or “Distribution of prime numbers and zeros of Dirichlet Lfunctions”. The twin themes of the course are to understand as well as possible the distribution of the zeros of Dirichlet Lfunctions (including the Riemann zetafunction), and then to use this knowledge to derive results on the distribution of prime numbers, with particular attention to their distribution within arithmetic progressions. The course will begin with a quick review of the prime number theorem and its analogue for arithmetic progressions.
Advertisement: There will be a conference on Lfunctions and their applications to number theory at the University of Calgary from May 30–June 3, 2011. Students who take this course should be wellprepared to get a lot out of that conference. Contact the instructor if you are interested in attending.
Prerequisites:
Students should have had a previous course in analytic number theory (for example, MATH 539 here at UBC). The background of students should include the following elements, all of which should be present in those who succeeded in MATH 539: a strong course in elementary number theory (for example, MATH 537), a graduate course in complex analysis (for example, MATH 508), and the usual undergraduate training in analysis (for example, MATH 320).
Evaluation: Each student will deliver three lectures for the course, and write up (in LaTeX) lecture notes corresponding to another student's three lectures. The lectures will be chosen by the student in consultation with the instructor from the list below; most students will choose to deliver consecutive lectures on the same topic.
The last day of classes is April 6, 2011; however, because there are no exams, the lectures will continue into the beginning of the final exams period to accommodate as many students and topics as possible.
Dates 
Speaker 
Topic 
Writer 
Draft due 
Article due 
Jan 10–12 
Greg 
Organization and introduction 



Jan 17–19 
Everyone 
Fourminute talks (all topics) 



Jan 21–28 
Greg 
Review on Lfunctions and primes in arithmetic progressions: explicit formula, zerofree region, exceptional zeros 



Jan 29–Feb 4 
Greg 
Primes in short intervals; irregularities of distribution (the Maier matrix method) 



Feb 7–11 
Nick 
Linnik's Theorem on the least prime in an arithmetic progression 
Colin 
Feb 21 
Feb 28 
Feb 14–18 

(no class) 



Feb 21–25 
Justin, Greg 
Zeros on the critical line 
Eric 
Mar 28 
Apr 4 
Feb 28–Mar 4 
Tatchai 
The large sieve and the Bombieri–Vinogradov Theorem 
Nick 
Mar 14 
Mar 21 
Mar 7–11 
Daniel 
The least quadratic nonresidue and the least primitive root modulo primes (unconditional and conditional results) 
Carmen 
Mar 21 
Mar 28 
Mar 14–18 
Eric 
Analytic number theory without zeros (current work of Granville/Soundararajan) 



Mar 21–25 
Li 
Oscillations of error terms, Littlewood's results 
Tatchai 
Apr 4 
Apr 11 
Mar 28–30 
Justin, Greg 
The nonvanishing of Lfunctions at the critical point and on the real axis 



Apr 4–8 
Carmen 
Limiting distributions of explicit formulas and prime number races 
Daniel 
Apr 18 
Apr 25 
Apr 11–15 
Colin 
The Selberg class of Lfunctions 
Li 
Apr 25 
May 2 
Apr 18 
Greg 
Horizontal distribution of zeros of Dirichlet Lfunctions; zerodensity theorems 



Apr 20 
Greg 
Proofs of the prime number theorem that avoid the zeros of ζ(s) 



References for these topics:
 H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory: I. Classical theory (some errata have been posted online)
Note: draft versions of some chapters from their future sequel Multiplicative Number Theory: II. Modern developments can be found on the web.
 H. Iwaniec and E. Kowalski, Analytic Number Theory
 E. C. Titchmarsh (revised by D. R. HeathBrown), The Theory of the Riemann ZetaFunction
 The primary research literature (you can find references in the above books or by speaking with the instructor), almost all of which is searchable at MathSciNet
Possible references for fundamental analytic number theory:
 H. Davenport, Multiplicative Number Theory
 A. E. Ingham, The Distribution of Prime Numbers
 T. M. Apostol, Introduction to Analytic Number Theory
 P. T. Bateman and H. G. Diamond, Analytic Number Theory: An introductory course
Possible references for elementary number theory:
 I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers
 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers
Use of the web: After the first day, no handouts will be distributed in class. All course materials will be posted on this course web page. All documents will be posted in PDF format and can be read with the free Acrobat reader. You may download the free Acrobat reader at no cost. You may access the course web page on any public terminal at UBC or via your own internet connection.
