Links:
L-functions
and Automorphic forms (Goldfeld Fest) (Conference photos)
Automorphic
Forms and L-Functions for the Group GL(n,R)
Joint
COLUMBIA-CUNY-NYU
Number Theory Seminar
Braid
Group
Cryptography
Bretton
Woods Workshop on Multiple Dirichlet Series(2005)
(Conference
photos)
Edinburgh
Workshop on Multiple Dirichlet Series(2008)
Decision Regarding
World
Record Musky Challenge
Very
accurate clock
Teaching, Spring 2009:
MATH W4007 Analytic Number
Theory 3 pts.
(Tuesday/Thursday 2:40-3:55pm, Room
520 Math Building)
TA: Min
Lee (Help Room Hours: Monday 11:00-12:00, Wednesday 11:00-12:00
and 3:00-4:00)
Course Description:
A one semeser course covering the theory of modular forms, zeta
functions, L -functions, and the Riemann hypothesis. Particular topics
covered include the Riemann zeta function, the prime number theorem,
Dirichlet characters, Dirichlet L-functions, Siegel zeros, prime number
theorem for arithmetic progressions, SL (2, Z) and subgroups, quotients
of the upper half-plane and cusps, modular forms, Fourier expansions of
modular forms, Hecke operators, L-functions of modular forms.
Homework:
#1: (due Tuesday, February 3) Let
G, G' be two cyclic groups which
contain
n, n' elements,
respectively. Construct all the characters of the direct product
G x
G' and show that these characters
themselves form a group which is isomorphic to
G x
G'. In the same manner explicitly
construct all possible Dirichlet characters (mod 35).
#2: (due Tuesday, February 10) (I) Let
χ
0 be the trivial character mod q, and let q
1 be
some factor of q. For any character χ
1 (mod q
1)
there is a character χ (mod q) defined by χ = χ
0 χ
1.
Express L(s, χ) in terms of L(s, χ
1). Conclude that L(1, χ)
= 0 if and only if L(1, χ
1) = 0.
(II) A character (mod q) that cannot
be obtained in this way
from any character mod a proper factor q
1 | q (a
factor other than q itself ) is called primitive. Show that any
Dirichlet character χ comes from a unique primitive character χ.
(III) Prove that if s has real part
σ > 1 then ζ (2σ)/ζ (σ) < |L(s, χ)| ≤ ζ (σ) for all
Dirichlet characters
χ.
#3: (due Tuesday, February
17) HW3.pdf
#4: (due Tuesday, February 24) Show
that Γ(1/2) = √π where Γ(s) is the Gamma
function. Show that Γ(s) has a simple pole at s = -5.
What is the residue? Use the previous two results
(together
with the functional equation of the Riemann zeta function ζ (s)
)
to evaluate ζ (−10).
#5: (due Tuesday, March 3) Assume
the prime number theorem. Using summation by parts, show that the
sum 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + . . . of
the recipocals of the primes diverges.
#6: (due Tuesday, March
10) HW6.pdf
#7: (due Tuesday, March
31) HW7.pdf
#8: (due Tuesday, April
7) HW8.pdf
#9: (due Tuesday, April
21) HW9.pdf
TAKE HOME FINAL EXAM (due Tuesday,
May 5 at 3:00 P.M.) FinalExam.pdf
...........
MATH G6660 Automorphic
Representations 4.5 pts.
(Tuesday/Thursday 11:00-12:15, Room
405 Math Building)
Course Description: This
course presents the Jacquet-Langlands theory
of automorphic representations for GL(n). The stress will be to develop
the theory with many concrete explicit examples. Local theory - principal
series
representations, special representations, (g, K)-modules, the
supercuspidal representations, tensor products of local
representations, L-functions, Godement-Jacquet proof of the analytic
continuation and functional equation of L-functions.
