MATH 775, SPRING 2014, PROBLEMS 1
Due 13th January
1. (The Larger Sieve of Gallagher [1971]) Let N be a subset of the integers in an
interval [M + 1, M + N ] and suppose card(N ) = Z . Let Z (q, h) denote the number
of n N such that n h (mod q ). For any
MATH 775, SPRING 2014, PROBLEMS 5
Due Monday 10th February
1. Suppose that R and x R with x 1. Prove that
(
n x
1
e(n) min x,
)
.
2. Suppose R, x, y R with x 1, y 1 and that there is an A R such that for
m y the complex numbers am satisfy |am | A.
(i) Pro
MATH 775, SPRING 2014, PROBLEMS 4
Due Monday 3rd February
1. (P. J. Cohen, oral communication 1977) Suppose that M and N are integers, N 1,
and that T (x) is a trigonometric polynomial as given in (19.8). Suppose that > 0 and
that the points xr are well s
Math 775, Spring 2014, Problems 3
Due Wednesday 29th January
1. Prove that if Q is a real number such that Q 1, then
1
log Q.
n
nQ
2. A natural number q is squarefree when it has no repeated prime factors, i.e.
(q )2 = 1. Let s(n) denote the squarefree k
Math 775 Analytic Number Theory II, Spring 2014, Problems 2
Due Wednesday 23rd January 2014
1. Let K , R N, M Z, N = KR + 1, xr = r/R (1 r R), an = 1 when n M + 1 (mod R)
and an = 0 otherwise. Show that
R
M +N
2
an e(nxr )
= (N 1 + 1/ )
r 1 n=M +1
M +N
|a
MATH 775, SPRING 2014, PROBLEMS 6
Due Monday 17th February
We suppose throughout that am and bn are complex numbers which satisfy
|am | log m,
|bn | 1
and that 1 u x, u M x/u and M M 2M . Let
SII =
am bn e(mn)
m>u u<nx/m
SII (M ) =
M <mM
and
unx/m
am bn
MATH 775, SPRING 2014, PROBLEMS 7
Due Monday 24th February
1. (Hooley (1972), Montgomery & Vaughan (1979) By lower and upper bound sifting
functions we mean functions : N R with the properties
m
m|n
(m)
m| n
+
m
m| n
respectively.
(i) Let + be an upper
MATH 775, SPRING 2014, SOLUTIONS 5
1. Suppose that R and x R with x 1. Prove that
Trivially LHS x. When Z, LHS =
e(x)1
e()1 e()
(
)
1
e(n) min x, .
n x
and so LHS
1
| sin |
1
2 .
2. Suppose R, x, y R with x 1, y 1 and that there an A R such that for m
i
MATH 775, SPRING 2014, SOLUTIONS 4
1. (P. J. Cohen, oral communication 1977) Suppose that M and N are integers, N 1, and that
T (x) is a trigonometric polynomial as given in (19.8). Suppose that > 0 and that the points xr
are well spaced in the sense that
Math 775, Spring 2014, Solutions 3
1
1. Prove that if Q is a real number such that Q 1, then
log Q.
n
nQ
x+y dz
min(n+1,Q) dz
1
For all x n and y [0, 1], n x
nQ n
z . Thus the sum is at least
z.
2. q is squarefree when (q )2 = 1. Let s(n) denote the sq
MATH 775, SPRING 2014, SOLUTION 1
1. (The Larger Sieve of Gallagher [1971]) Let N be a subset of Z of the integers in an interval [M + 1, M + N ], and let Z (q, h) denote the number of n N
such that n h (mod q ). For any primepower q let r(q ) denote the
Math 775, Spring 2014, Solutions 2
1. Let K , R N, M Z, N = KR + 1, xr = r/R (1 r R), an = 1 when n M + 1 (mod R) and an = 0
otherwise. Show that
2
R
M +N
M +N
an e(nxr ) = (N 1 + 1/ )
|a n |2
r 1 n=M +1
n=M +1
where = min xr xs .
r =s
K
The sum over n o