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PMATH 740
Analytic Number Theory
Instructor: Stephen New
Term: Spring 2012 (1125)
University of Waterloo
July 30, 2012
Disclaimer: These notes are provided as-is, and may be incomplete or contain errors.
Contents
1

PMATH 740 Analytic Number Theory, Assignment 5
Not to hand in
1: Let l Z+ and let 1 denote the identity character in Ul .
1
1 pz .
(a) Show that L1 (z ) = (z )
1
p|l
(b) We can use part (a) to extend L1 (z ) to be holomorphic in C \ cfw_1 with a simple po

PMATH 740 Analytic Number Theory, Assignment 4
1: (a) Show that
p x
Due Fri July 20
1
converges.
p log p
(b) Determine whether
n=pk x
1
converges.
n (n)
2: (a) Let : Z C be periodic, completely multiplicative and not identically zero. Let l
be the smalles

PMATH 740 Analytic Number Theory, Assignment 3
log n
=
n
nx
(n)
(b) Show that
=
n
1: (a) Show that
1
2
log2 x + c + O
1
2
Due Fri June 22
log2 x + 2 log x + O(1).
log x
x
for some constant c.
nx
(c) Show that for 1 < a R we have
nx
log x
(n)
=
+ (a)2 +

PMATH 740 Analytic Number Theory, Assignment 2
Due Fri June 8
1: (a) Find the number of cubes, and the number of twelfth powers in U81 .
(b) Find the number of cubes, and the number of twelfth powers in U128 .
(c) For n = 18900, nd the universal exponent

PMATH 740 Analytic Number Theory, Assignment 1
Due Fri May 25
1: Let a = (25)! and b = (5500)3 (1001)2 .
(a) Find the prime factorization of a and of b.
(b) Find the prime factorization of gcd(a, b) and of lcm(a, b).
(c) Find the number of positive factor