Math 892: (Winter Semester)
Core Course in Analysis, II
Text: W. Rudin, Real and Complex Analysis, 3rd
Edition, McGraw Hill, 1987
Time:Mon. 12noon1.30pm; Thu. 1.00pm2.30pm; Jeff 422.
Assignment 1
Assignment 2
Assignment 3
Recent Books






Problems in the Theory of Modular Forms 
Transcendental Numbers 
Indian Philosophy:
An Introduction 
The Mathematical Legacy of Srinivasa Ramanujan 
A First Course in Graph Theory and Combinatorics 
Automorphic Lfunctions 
(with M. Dewar & H. Graves), IMSc Lecture Notes, 2015 
(with P. Rath), Springer, 2014. 
Broadview Press, 2013. 
(with V. Kumar Murty), Springer, 2012. errata 
(with S. Cioaba), HBA, 2009
errata 
(with J. Cogdell and H. Kim), Fields Inst./AMS, 2009. 
Earlier Books
Useful Links
Number Theory Seminar & Infosheet
Mathscinet Search
Number Theory Web
Kumar Murty's Page
Library/Queen's University
JSTOR(Mathematics)
Some vintage photos
Research Interests
Number Theory: Artin's conjecture, Elliptic curves, Modular Forms,
Automorphic forms, Langlands Program, Selberg's conjectures, Sieve Methods,
Cryptography
More research description can be found at
CICMA
Here is chronological list of mathematical publications:
LIST
Here is the list of philosophical publications: LIST
This is the link for Mathematical Reviews of my papers.
Here is the link to my mathematical genealogy.
Degrees and Honours
 Ph.D. (MIT, 1980) (Advisor: Harold Stark)
 CoxeterJames Prize, 1988
 Fellow of the Royal Society of Canada (1990  )
 E.W.R. Steacie Fellow (19911993)
 J.H.H. Nesbitt Lecturer, 1992
 Kenneth Ireland Lecturer, 1993
 Balaguer Prize (1996) jointly with V. Kumar Murty (see August 1996 issue of the Notices of the AMS)
 Killam Research Fellow (19982000)
 JefferyWilliams Prize, 2003
 Queen's Research Chair, 2002
 Fellow of the Fields Institute, 2003
 Queen's Research Prize, 2003
 Adjunct Professor, McGill University (Montreal), HarishChandra Research Institute (Allahabad), Institute for Mathematical Sciences (Chennai), Chennai Mathematical Institute (Chennai), Indian Institute of Technology (Mumbai), Tata Institute for Fundamental Research (TIFR, Mumbai).
 Crossappointment to the Department of Philosophy
 Elected fellow of the National Academy of Sciences (India), 2007.
 Elected fellow of the Indian National Science Academy (INSA), 2008.
 Elected fellow of the American Mathematical Society, 2012.
 Simons Fellowship, 20132014.
Expository Lectures
The Art of Research, 2005 Herzberg Lecture, Carleton University/Tata Institute for Fundamental Research, Mumbai, India.
Ellipses and elliptic curves, ColemanEllis Lecture, March 30, 2006.
Recent Developments in the Langlands Program, C.R. Math. Acad. Sci. R. Can., 24(2002), no. 2, 3354.
A Motivated Introduction to the Langlands Program, in Advances in Number Theory, (edited by F. Gouvea and N. Yui), pp. 3766,
Clarendon Press, Oxford, 1993.
Ramanujan graphs, Journal of the Ramanujan Math. Society, 18, No. 1, (2003), 120.
Selberg's conjectures and Artin Lfunctions, Bulletin of the American Math. Society, 31(1) (1994), 114.
Recent Publications
Sudoku squares and chromatic polynomials(with A. Herzberg), Notices of the Amer. Math. Society, 54(2007), no. 6, 708717.
Multiple Hurwitz zeta functions (with K. Sinha), in Multiple Dirichlet series, automorphic forms, and analytic number theory, Proc. Symp. Pure Math., 75 (2006), 135156.
An introduction to sieve methods and their applications, (with A. Cojocaru) London Mathematical Society Student Texts, 66, Cambridge University Press, 2006.
Problems in Analytic Number Theory, Graduate Texts in Mathematics, Vol. 206, SpringerVerlag, 2001.
Lectures on Symmetric Power Lfunctions (Fields Institute, 2004)(pdf format)
Problems in Algebraic Number Theory, (with J. Esmonde), Graduate Texts in Mathematics, 190, SpringerVerlag, 2005, 2nd edition.
Nonvanishing of Lfunctions and Applications, Progress in Mathematics 157, Birkhauser, 1997 (Balaguer Prize Monograph), (with Vijaya Kumar Murty)
Spectral Estimates for Abelian Cayley Graphs,
(with J. Friedman and JP. Tillich), Journal of Combinatorial Theory, Series
B., 96(2006), no. 1, 111121.
The square sieve and the LangTrotter conjecture (with A. Cojocaru and E. Fouvry), Canadian Journal of Math., 57(6) (2005), 11551177.
Sieve methods in combinatorics (with Y.R. Liu), Journal of Combinatorial Theory, Series A, 111(2005), no. 1, 123.
On decimations of lsequences(with M. Goresky, A. Klapper, and I. Shparinski), SIAM Journal of Discrete Mathematics, 18 (2004), no. 1, 130140.
Euclidean rings of algebraic integers (with M. Harper), Canadian Journal of Mathematics, 56(2004), no. 1, 7176.
Cyclicity of elliptic curves modulo p and elliptic analogues of Linnik's problem, (with A. Cojocaru), Math. Annalen, 330 (2004), no. 3, 601625.
On a conjecture of Birch and SwinnertonDyer (with W. Kuo), Canadian Journal of Math., 57(2005), no. 2, 328337.
An application of Mumford's gap principle, (with J.J. Lee), Journal of Number Theory, 105(2004), no. 2, 333343.
Current PostDoctoral Fellows:Kevser Aktas, Arunabha Biswas, Shuntaro Yamagishi.
Theses supervised:
 Ian Miyamoto, Primes and elliptic curves, M.Sc. thesis, McGill University, 1988.
 David Clark, Lseries and ranks of elliptic curves, M.Sc. thesis, McGill University, 1988. (clark@math.byu.edu)
 David Clark, The Euclidean algorithm for Galois extensions of the rational numbers, Ph.D. thesis, McGill University, 1992.(clark@math.byu.edu)
 Tomasz Stefanicki, Nonvanishing of Lfunctions attached to automorphic representations of GL(2), Ph.D. thesis, McGill University, 1992.
 Francesco Pappalardi, On Artin's conjecture for primitive roots, Ph.D. thesis, McGill University, 1993. (pappa@mat.uniroma3.it)
 Chantal David, Supersingular Drinfeld modules, Ph.D. thesis, McGill University, 1993. (chantal@discrete.concordia.ca)
 Ashwaq Hashim, Application of Lindelof hypothesis to cusp forms, M.Sc. thesis, McGill University, 1993. (ucakash@ucl.ac.uk)
 Yuanli Zhang, Some analytic properties of automorphic Lfunctions, Ph.D. thesis, McGill University, 1994.(yz@math.purdue.edu)
 Sridhar Narayanan, Selberg's conjectures, M.Sc. thesis, McGill University, 1994.(sri@esankya.com)
 Keldon Drudge, The ABC Conjecture, M.Sc. Thesis, McGill University, 1995.
 Minatsu Yamaji, Applications of Brauer Induction to Artin Lfunctions, M.Sc. Thesis, McGill University, 1996.
 Francesco Sica, The order of vanishing of Lfunctions at the center of the critical strip, Ph.D. thesis, McGill University, 1998.
 Satya Mohit, The Wieferich criterion, the ABC conjecture and Shimura's correspondence, M.Sc. thesis, Queen's University, 1998.
 YuRu Liu, The Turan sieve and some of its applications, M.Sc. thesis,
Queen's University, 1998.(yrliu@math.harvard.edu)
 Filip Saidak, On conjectures about the distribution of primes in various sequences,
M.Sc. thesis, Queen's University, 1998.(filips@math.ucalgary.ca)
 Malcolm Harper (PhD, McGill), A proof that $Z[\sqrt{14}]$ is Euclidean, Ph.D. thesis, McGill University,
2000.
 Steven Pope,
On Problems of Cryptography: Primality testing and Factoring integers, M.Sc.
thesis, Queen's University, 2000.
 Filip Saidak,
On nonabelian generalizatons of the ErdösKac theorem, Ph.D. thesis, Queen's University, 2001.
 Alina Cojocaru,
Cyclicity of Elliptic Curves modulo p, Ph.D. thesis, Queen's University, 2002.
 Kaneenika Sinha, Generalized Hurwitz zeta functions, (MSc, Queen's, 2002).
 Talia Borodin, Primality Testing and Factoring, (M.Sc., Queen's, 2003).
 Benjamin Ju, Elementary Proofs of the Prime Number Theorem,
(M.Sc., Queen's, 2003).
 Tim Wotherspoon, Analysis of the NarayanaFermat Factoring Algorithm(M.Sc., Queens, 2004).
 Jeanine Van Order, Counting Integral Ideals in an Algebraic Number Field,(M.Sc., Queen's, 2005).
 Sebastian Cioaba, Eigenvalues, Expanders and Gaps between Primes, (Ph.D., Queen's, 2005).
 Kaneenika Sinha, Effective Equidistribution of Eigenvalues of Hecke Operators, (Ph.D., Queen's, 2006).
 Brandon Fodden, Diophantine equations and the Generalized Riemann hypothesis (Ph.D., Queen's, 2007).
 Marilyn Reece Myers, Variations of Kummer's Conjecture (Ph.D., Queen's, 2007).
 Chester Weatherby, Transcendence of various infinite series and applications of Baker's theorem (Ph.D., Queen's, 2009).
 Robert Rundle, Generalized Pythagorean Triples (M.Sc., Queen's, 2007).
 Adam Felix, The order of a mod p and Artin's conjecture(M.Sc., Queen's, 2007).
 Andrew Droll, Properties of Ramanujan graphs (M.Sc., Queen's, 2008).
 Marc Papais, Graph spectra and the Ihara zeta function: implications and connections (M.Sc., Queen's, 2008).
 Carpio Cheng, Grimm's conjecture and a Gaussian analogue, (M.Sc. Queen's, 2009).
 Adam Felix,, Variations on Artin's primitive root
conjecture (Ph.D., Queen's, 2011)
 Robert Rundle, Generalization of Ruderman's problem to imaginary quadratic fields (Ph.D., Queen's, 2012)
 Andrew Droll,, Variations of Li's criterion for an extension of the Selberg class (Ph.D., Queen's, 2012).
 Ryan Case, RamanujanFourier series and applications (M.Sc., Queen's, 2013).
 Hector Pasten, Arithmetic problems around the abc conjecture
and connections with logic, (Ph.D., Queen's, 2014).
 Siddhi Pathak, The Chowla problem and its generalizations, (M.Sc., Queen's, 2015)
 Kannappan Sampath , The saddle point method and its applications to number theory(M.Sc., Queen's, 2015)
 Akshaa Vatwani, Higher rank sieves and applications(Ph.D., Queen's, in progress).
 PengJie Wong (Ph.D., Queen's, in progress).
 Francois Seguin (Ph.D., Queen's, in progress).
 Siddhi Pathak, (Ph.D., Queen's, in progress).
 Kannappan Sampath (Ph.D., Queen's, in progress).
 Arpita Kar (M.Sc. Queen's, in progress).
