18.781 ANALYTIC NUMBER THEORY QUIZ
Friday, Nov. 30, 2007
Name:
Numeric Student ID:
Instructor’s Name:
I agree to abide by the terms of the honor code:
Signature:
Instructions:
Print your name, student ID number and instructor’s name in the space
provided.
During the test you may not use notes, books or calculators.
Read each
question carefully and
show all your work
; full credit cannot be obtained without
sufficient justification for your answer unless explicitly stated otherwise. Underline your
final answer to each question. There are 4 questions. You have 50 minutes to do all the
problems.
Question
Score
Maximum
1
6
2
6
3
6
4(BONUS)
3
Total
18
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Question 1 of 4, Page 2 of 5
Solutions
1. Give an explicit definition of the characters
χ
q
: (
Z
/q
Z
)
×
→
C
×
, where
q
is
prime. That is, provide a careful definition of such a function
χ
q
and then prove
that your definition satisfies the necessary property of a character.
Solution:
Everything on this quiz can be found someplace in the notes posted online, so I
will be relatively brief on these solutions.
The character
χ
q
to the modulus
q
is defined by
χ
q
(
n
) =
ζ
ν
(
n
)
q

1
where
ζ
q

1
is a (
q

1)st root of unity, and
ν
(
n
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 Spring '09
 BRUBAKER
 Number Theory, Arithmetic progression, Prime number, Riemann zeta function, primitive root mod

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