MATH 571 ANALYTIC NUMBER
THEORY I, SPRING 2011, PROBLEMS 1
Due 18th January
1. Solve where possible
(i) 77x 84 (mod 143);
(ii) 77x 84 (mod 147);
2. Find the last digit of 72001 and of 13143 .
3. Suppose that f : N Z is a totally multiplicative function wi
THE BOMBIERI-VINOGRADOV THEOREM
R. C. Vaughan
1. The Main Theorem
The Bombieri-A. I. Vinogradov Theorem is concerned with the distribution of
primes into arithmetic progressions. By the way, the other Vinogradov, I. M.,
will also make an appearance, albei
Warings Problem: A Survey
R. C. Vaughan1 and T. D. Wooley2
1
The Classical Waring Problem
Omnis integer numerus vel est cubus, vel e duobus, tribus, 4, 5, 6, 7,
8, vel novem cubis compositus, est etiam quadrato-quadratus vel e duobus,
tribus, &c. usque ad
MATH 571 ANALYTIC NUMBER
THEORY I, FALL 2012, PROBLEMS 3
Due 18th September
1. Let q N, n = (q ), g1 , . . . , gn be the elements of Z in some order, 1 , . . . , n
q
be the characters modulo q in some order, and uij = i (gj )/ (q ), U = (uij )nn .
Prove t
MATH 571 ANALYTIC NUMBER
THEORY I, FALL 2012, PROBLEMS 4
Due 25th September
1. Using only results proved in class show that if is a Dirichlet character modulo
q , then | ()| q .
2. (i) Show that
1
(q )
(a) () =
e(a/q )
(a, q ) = 1,
0
otherwise.
(ii) Show
Math 571 Analytic Number Theory I, Fall 2012, Problems 9
Due Thursday 8th November 2012
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
2
M +N
an e(nxr )
M +N
|an |2
= (N 1 + 1/ )
r 1 n=M +1
MATH 571 ANALYTIC NUMBER
THEORY I, FALL 2012, PROBLEMS 7
Due 23rd October
1
We use C0 to denote Eulers constant, so that for every x 1, nx n = log x +
s
1
1
s
C0 + O( x ). We recall that (s) is dened by (s) = seC0 s n=1 (1 + n )e n
1
and that the entire f
MATH 571 ANALYTIC NUMBER
THEORY I, FALL 2012, PROBLEMS 6
Due 9th October
1. (Plya 1918, Vinogradov 1918) Suppose that q > 1, let be a character modulo
o
q and for M Z, n N dene
M +N
S (M, N, ) =
(n).
n=M +1
(i) Suppose that is primitive. Prove that
q
N
1
Math 571 Analytic Number Theory I, Spring 2011, Problems 12
Due Tuesday 12th April
q
ank
Given positive integers k and q dene Gk (q, a) =
e
and let Nk (q, h) denote the number
q
n=1
of solutions of the congruence xk h (mod q ). p denotes a prime number.
Math 571 Analytic Number Theory I, Spring 2011, Problems 11
Due Tuesday 5th April
1. Let 2 (q ) denote the number of primitive characters (mod q ).
(a) Show that 2 (q ) is a multiplicative function.
(b) Show that d|q 2 (d) = (q ).
(c) Show that
2
1 2
2 (
Math 571 Analytic Number Theory I, Spring 2011, Problems 10
Due Tuesday 29th March
1. (a) Show that for arbitrary real or complex numbers c1 , . . . , cq ,
q
q
2
c n ( n ) = ( q )
|cn |2
n=1
n=1
(n,q )=1
where the sum on the left hand side runs over all D
MATH 571 ANALYTIC NUMBER
THEORY I, SPRING 2011, PROBLEMS 2
Due 25th January
1. Prove that if f is a multiplicative function, then so is g , dened by
g ( n) =
f ( m) .
m |n
2. Prove that
(m) =
m |n
1
n = 1,
0
n > 1.
3. Prove that (n) = n m|n (m) .
m
4. Pro
MATH 571 ANALYTIC NUMBER
THEORY I, SPRING 2011, PROBLEMS 3
Due 1st February
1. (i) Let m N. Prove that
(y 1)(y m1 + y m2 + + y + 1) = y m 1.
(ii) Let n N. Prove that
(x2 + 1)(x2 1)(x4n4 + x4n8 + + x4 + 1) = x4n 1.
(iii) Let p be a prime number with p 1 (m
MATH 571 ANALYTIC NUMBER
THEORY I, SPRING 2011, PROBLEMS 4
Due 8th February
1. (H.-E. Richert, unpublished) (a) Show that
x<nx+y
d 2 |n
d
2
2
d e
=y
+O
| d |
.
[d, e]2
d, e
d
(b) Let f (n) = n2 p|n 1 p2 . Show that d|n f (d) = n2 .
(c) For 1 d z let d
MATH 571 ANALYTIC NUMBER
THEORY I, SPRING 2011, PROBLEMS 5
Due 15th February
1. (van Lint & Richert (1965) (a) Show that
( n) 2
( d) 2
( n )
( d )
n z
(b) Deduce that
m z
(m,q )=1
d |q
(m)2
.
(m)
( n) 2
( q ) ( n ) 2
.
( n )
q
( n )
n z
(n,q )=1
n z
2
MATH 571 ANALYTIC NUMBER
THEORY I, SPRING 2011, PROBLEMS 6
Due 22nd February
1. Let 2 (x) denote the number of primes p x such that p + 2 is prime. Show
that
x
1
2 ( x )
2 ( t )
=
+
dt
p
x
t2
2
px
p+2 prime
and deduce that
1
p
p
p+2 prime
converges.
2. Le
Math 571 Analytic Number Theory I, Spring 2011, Problems 7
Due Tuesday 1st March
1. (Vaughan 1973). Suppose that for each prime p, 0 b(p) < p and dene L(Q) =
(a) Prove that L(Q) =
( q ) 2
q Q
p |q
b( p)
.
p b( p)
b( p) h
where s(r) is the squarefree kern
Math 571 Analytic Number Theory I, Spring 2011, Problems 8
Due Tuesday 15th March
1. (Gallagher 1967.) (a) Suppose that > 0 and f is continuous on [, ]. By considering
0
f ( )d and 0 f ( )d separately and integrating by parts, or otherwise, prove that
2f
Math 571 Analytic Number Theory I, Spring 2011, Problems 9
Due Tuesday 22nd March
1. (Carmichael (1932) Let c(q ; n) be Ramanujans sum, as dened in homework 6. (a) Show
that if q > 1, then
q
c(q ; n) = 0.
n=1
(b) Show that if q1 = q2 and [q1 , q2 ]|N , th
8. Dirichlets Theorem and Farey Fractions We are concerned here with the approximation of real numbers by rational numbers, generalizations of this concept and various applications to problems in number theory. A property of the integers which we frequent