analytic methods in number theory

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Analytic Methods in Number TheoryAnand Jyoti SanasamEntry Number: 2011MAS7137

April 25, 2013

Contents1 Introduction 52 Arithmetic Functions And Dirichlet Multiplication 6

2.1 The Ring Of Arithmetic Functions . . . . . . . . . . . . . . . . . . .

. . . . . 6

2.2 Mean Values of Arithmetic Functions . . . . . . . . . . . . . . . . . .

. . . . . 7

2.3 Multiplicative Functions and The M¨obius Function . . . . . . . . . . .. . . . 8

2.4 The Mean Value of Euler Phi Function . . . . . . . . . . . . . . . . .

. . . . . 10

3 Divisor Functions 11

3.1 The Sum and Number of Divisors . . . . . . . . . . . . . . . . . . . .

. . . . . 11

3.2 Perfect Number . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . 12

4 An Introduction To The Prime Number Theorem 14

4.1 The Mangoldt Function Λ(n ) . . . . . . . . . . . . . . . . . . . . . .. . . . . 14

4.2 Chebyshev’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . 15

4.3 The Prime Number Theorem . . . . . . . . . . . . . . . . . . . . . . .

. . . . 16

5 Characters of A Finite Abelian Group And Primes in Arithmetic Progressions17

5.1 Characters of Finite Abelian Groups . . . . . . . . . . . . . . . . . .

. . . . . 17

5.2 Dirichlet L-Functions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 18

5.3 Dirichlet’s Theorem for Primes of the Form 4n − 1 and 4n + 1 . . . . .

. . . . 21

6 Introduction to Additive Number Theory 22

6.1 Sum of Four Squares . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . 22

7 Waring’s Problem 24

7.1 Stable Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . 24



7.2 Shnirel’man’s Theorem . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . 25

7.3 Waring’s Problem for Polynomials . . . . . . . . . . . . . . . . . . .

. . . . . . 28

8 Sums of Sequences of Polynomials 32

8.1 Sums and Differences of Weighted Sets . . . . . . . . . . . . . . . . .

. . . . . 32

8.2 Linear and Quadratic Equations . . . . . . . . . . . . . . . . . . . .

. . . . . . 35

Bibliography 36

3

Chapter 1

IntroductionIn mathematics, “Analytic Number Theory” is a branch of number theory

that uses

methods from mathematical analysis to solve various problems in number

theory. It can be

divided into two important parts, namely Multiplicative Number Theory and

Additive

Number Theory.

Multiplicative number theorem contains about divisors, primes, distribution

of primes andmany theorems and conjectures including Erd¨os-Selberg elementary proof of

prime number

theorem.

On the other hand, additive number theory consists of problems in partition

functions,

Waring’s problem for polynomials, Riemann Hypothesis, representation of

numbers in different

forms like Liouville’s method etc.

In the first few chapters, I look into various prospects of arithmetic

functions including

divisor functions, M¨obius functions etc. and their properties. Then I look

for the distribution

of primes. A chapter on the introduction of the prime number theorem with

some proofs has

been added. A chapter is devoted to the Dirichlet characters and their

roles in proving some

theorems on the prime numbers in arithmetic progressions. Again, a brief

note for the future

work is also given.5



Chapter 2Arithmetic Functions And Dirichlet

Multiplication2.1 The Ring Of Arithmetic FunctionsDefinition 2.1.1. Ari thmet ic Funct ion

An arithmetic function is a complex-valued function whose domain is the set of positive integers.Definition 2.1.2. Summat ion Funct ion

Let f (n ) and g (n ) be functions defined on the set of positive integers. We say g (n )

is the

summation function of f (n ) if

g (n ) =

Σ d | n

f (d ). (2.1)

We define two basic binary operations as follows:

• The pointwise sum f + g of the arithmetic functions f and g is defined by

(f + g )(n ) = f (n ) + g (n ) (2.2)

• The Dirichlet convolution f ∗ g is defined by

(f ∗ g )(n ) =

Σ d | n

f (d )g (n/d ) (2.3)

Theorem 2.1.1. The set of all complex-valued arithmetic functions, with pointwiseaddition

and Dirichlet multiplication is a commutative ring with additive identity 0(n ) and

multiplicative

identity I (n ).

Corollary: If f is an arithmetic function such that f (1) = 0, then f − 1 exists

and is given by

f − 1(1) = 1

f (1) ,

f − 1(n ) = − 1

f (1)

Σ d | n

f (n/d )f − 1(d ). (2.4)

6

2.2 Mean Values of Arithmetic FunctionsDefinition 2.2.1. Mean Values of Ari thmet ic Funct ion s

We define the mean value F (x ) of an arithmetic function f (n ) by



F (x ) =

Σ n≤ x

f (n ) (2.5)

where the sum is all positive integersn ≤

x .

Theorem 2.2.1. Let a and b be integers with a < b , and let f (t ) be a function that

is

monotonic on the interval [a, b ]. Then

min

(

f (a ), f (b )

)

≤

Σb

n=a

f (n ) −

∫ b

a

f (t )dt ≤ max

(

f (a ), f (b )

)

. (2.6)

Let x and y be real numbers with y < [x ], and let f(t) be a nonnegative monotonic

functionon [y, x ]. Then

_____ Σ y<n≤ x

f (n ) −

∫ x

y

f (t )dt

_____

≤ max(

f (y ), f (x )

)

. (2.7)

Proof. If f (t ) is increasing on [n, n + 1], then

f (n ) ≤

∫ n+1

n

f (t )dt ≤ f (n + 1).

If f (t ) is increasing on the interval [a, b ], then



f (a ) +

∫ b

a

f (t )dt ≤

Σb

n=a

f (n ) ≤ f (b ) +

∫ b

a

f (t )dt.

Similarly, if f (t ) is decreasing on [n, n + 1], then

f (n + 1) ≤

∫ n+1

n

f (t )dt ≤ f (n ).

If f (t ) is decreasing on the interval [a, b ], then

f (b ) +

∫ b

a

f (t )dt ≤

Σb

n=a

f (n ) ≤ f (a ) +

∫ b

a

f (t )dt.This proves (2.6).

Let f (t ) be nonnegative and monotonic on the interval [y, x ]. Let a = [y ] +

1 and b = [x ].

Thus y < a ≤ b ≤ x . If f (t ) is increasing, then

Σ y<n≤ x

f (n ) =

Σ a<n≤ b

f (n ) (2.8)

≤

∫ b

a

f (t )dt + f (b ) (2.9)

≤

∫ y

x

f (t )dt + f (x ). (2.10)

Since

f (a ) ≥



∫ a

y

f (t )dt

and

f (x ) ≥

∫ x

b

f (t )dt,

it follows that Σ y<n≤ x

f (n ) ≥

∫ b

a

f (t )dt + f (a ) ≥

∫ y

x

f (t )dt − f (x ).

Therefore, _____

Σ y<n≤ x

f (n ) −

∫ x

y

f (t )dt

_____

≤ f (x ).

If f (t ) is decreasing, then by a similar calculation, we get

_____ Σ y<n≤ x

f (n ) −

∫ x

y

f (t )dt

_____

≤ f (y ).

2.3 Multiplicative Functions and The M¨obius FunctionDefinition 2.3.1. Mult ip l icat ive func t ion

If f is an arithmetic function such that

f (mn ) = f (n )f (m ) (2.11)

whenever (m, n ) = 1, then we say that f is a multiplicative function. And if we

remove the

restriction (m, n ) = 1, we say that f is totally multiplicative.

Theorem 2.3.1. If f is a multiplicative function and F be defined by

F (n ) =Σ



d | n

f (d ), (2.12)

then F is also multiplicative.

Proof. Let m and n be relatively prime integers. Then

F (mn ) =

Σ d | n

f (d ) =

Σ d 1| m;d 2| n

f (d 1d 2).

Since every divisor d of mn can be uniquely written as a product of divisor

d 1 of m and a

divisor d 2 of n , where (d 1, d 2) = 1. Again, f is multiplicative, therefore

f (d 1d 2) = f (d 1)f (d 2).

Thus,

F (mn ) =

Σ d 1| m;d 2| n

f (d 1)f (d 2) (2.13)

=

Σ d 1| m

f (d 1)

Σ d 2| m

f (d 2)

(2.14)

= F (m )F (n ). (2.15)Theorem 2.3.2. Let h = f ∗ g . If g and h are both multiplicative, and neither g nor h

is the

zero function, then f is also multiplicative.

Definition 2.3.2. The M •obius Funct ion The M¨obius Functionμ(n ) is defined as follows:

μ(n ) =

1 if n = 1;

(− 1)k if n is the product of k distinct primes;



0 if n is divisible by the square of a prime.

Theorem 2.3.3. The M¨obius function is multiplicative, and Σ d | n

μ(d ) =

{1 if n = 1;

0 if n > 1.

Theorem 2.3.4. M¨obius Inversion Formula

g (n ) =

Σ d | n

f (d ) (2.16)

if and only if

f (n ) =

Σ d | n

g (d )μ(n/d ). (2.17)

Proof. It suffices to show that g = f ∗ 1 if and only if f = g ∗ μ. If g = f

∗ 1,

g ∗ μ = (f ∗ 1) ∗ μ = f ∗ (1 ∗ μ) = f ∗ I = f

where I (n ) is defined as

I (n ) =

{

1 if n = 1;

0 if n > 1.

Converse is similar.

Theorem 2.3.5. Let f be a multiplicative function. Then

Σ d | n

μ(d )f (d ) =

Π p| n

(

1 − f ( p )

)

. (2.18)

2.4 The Mean Value of Euler Phi FunctionDefinition 2.4.1. Euler’s Totient Function

Let ϕ(n ) denote the number of integers, k , such that 1 ≤ k ≤ n and (k, n ) = 1.

Then we call

ϕ(n ) the Euler’s totient function or Euler’s phi function.

Theorem 2.4.1. For x ≥ 1,

Φ(x ) =



Σ n≤ x

ϕ(n ) =

3x 2

π2 + O (x log x ). (2.19)

Proof.

Φ(x ) =

Σ n≤ x

ϕ(n ) (2.20)

=

Σ n≤ x

Σ dd ′ =n

d ′ μ(d ) (2.21)=

1

2

Σ n≤ x

μ(d )

[x

d

] ([x

d ]

+ 1

)

. (2.22)

This implies that

Φ(x ) =

1

2

Σ n≤ x

μ(d )

((x

d

)2

+ O

(x

d

))

(2.23)

=



x 2

2

Σ n≤ x

μ(d )

d 2 + O

(

x

Σ d ≤ x

1

d

)

(2.24)

=

3x 2

π2 + O (x log x ). (2.25)

(2.26)

Theorem 2.4.2. The probability that the two positive integers are relatively prime is

6/ π2.

Chapter 3Divisor Functions

3.1 The Sum and Number of DivisorsDefinition 3.1.1. Sum of Div isors Funct ion

The sum of the divisors of a positive integer n is the sum of all positive divisors of n

and it isgiven by

σ(n ) =

Σ d | n

d. (3.1)

Definition 3.1.2. Number of Div isors Funct ion

The number of divisors of a positive integer n is the number of positive divisors of n

and isgiven by

τ (n ) =

Σ d | n

1. (3.2)

Theorem 3.1.1. The arithmetic functionsτ and σ are multiplicative, i.e.,

τ (mn ) = τ (m )τ (n ) (3.3)

and

σ(mn ) = σ(m )σ(n ) (3.4)

whenever m and n are relatively prime integers.



Corollary 3.1.1. If n =

Πr

i =1 p ei

i , where p i are distinct primes, then

τ (n ) =

Πr

i =1

(e i + 1). (3.5)

11

Corollary 3.1.2. If n =

Πr

i =1 p ei

i , where p i are distinct primes, then

σ(n ) =

Πr

i =1

p ei +1

i

− 1

p i − 1

. (3.6)

Theorem 3.1.2. Let f (n ) be multiplicative function. If

lim pk →∞

f ( p k ) = 0 (3.7)

as p k runs through the sequence of all prime powers, then

limn→∞

f (n ) = 0. (3.8)

Theorem 3.1.3. For every ε > 0,

τ (n ) ≪ " n " . (3.9)

i.e., the function f (n ) = τ (n )/n " → 0 as n → ∞ ,

where ≪ means that if f ≪ g then f = O (g ).

Proof. Let ε > 0. The function f (n ) = d (n )/n " is multiplicative. Therefore,it suffice to show

that

limn→∞

f (n ) = 0.

for every prime p . We know that

k + 1

2k"=2

is bounded for k ≥ 1, and so

f ( p k ) =



d ( p k )

p k" =

k + 1

2k" =

(

k + 1

p k"=2

)(

1

p k"=2

)

⇒ f ( p k ) ≤

(

k + 1

2k"=2

)(

1

p k"=2

)

≪

(

1

p k"=2

)

.

Hence,

limn→∞

f (n ) = 0.

3.2 Perfect Number Definition 3.2.1. Perfect Num ber

A number n is said to be perfect if it is equal to the sum of all positive divisors of n

less than

n . In short,

n is perfect if

σ(n ) = 2n .Theorem 3.2.1. Euclid-Euler

Let n be an even number. Then n is perfect if and only if

n = 2 p− 1(2 p − 1) (3.10)

where 2 p − 1 is a prime.

Definition 3.2.2. Mersenne Prime

If q is a prime such that q = 2 p− 1, we say that q is a Mersenne prime.

• 243112609− 1 is currently the largest Mersenne prime. It has 12, 978, 189

decimal digits.

It was discovered on August 23, 2008 by Edson Smith.



Chapter 4An Introduction To The Prime

Number TheoremWe introduce π(x ) to denote the number of primes not exceeding x , i.e.,

π(x ) =

Σ p≤ x

1 (4.1)

where p runs over all primes less than or equal to x .

Theorem 4.0.2. There are infinitely many primes.

4.1 The Mangoldt Function _(n )

Here we introduce Mangoldt functionΛ

which plays a very important role inthe distribution

of primes.

Definition: For every n ≥ 1 we define

Λ(n ) =

{

log p if n = p m for some prime p and some m ≥ 1;

0 otherwise.

Theorem 4.1.1. If n ≥ 1 we have

log n =Σ d | n

Λ(d ). (4.2)

and

Λ(n ) =

Σ d | n

μ(d ) log(n/d ) = −

Σ d | n

μ(d ) log d. (4.3)

Proof. The theorem is certainly true for n = 1. Let n > 1 and let

n =

Πr

k =1

p ak

k .

14

Taking logarithms on both side, we get

log n =Σr



k =1

a k log p k .

Now consider the right side of (4.4). The only nonzero terms in the sum

come from those

divisors d of the form p ak

k for m = 1, 2, ....a k . HenceΣ d | n

Λ(d ) =

Σr

k =1

Σak

m=1

Λ( p m

k ) (4.4)

=

Σr

k =1

Σak

m=1

log p k (4.5)

=

Σr

k =1

a k log p k (4.6)

= log n. (4.7)

Finally, inverting (4.4) by the M¨obius Inversion Formula, we get the

desired second result.

4.2 Chebyshev’s Theorem Again, we introduce functions ϑ (x ), called Chebyshev's ϑ funct ion and ψ

(x ), called

Chebyshev's ψ funct ion defined respectively by

ϑ (x ) =

Σ p≤ x

log p = log

Π p≤ x

p (4.8)

where p runs over all primes ≤ x , and

ψ(x ) =

Σ pk ≤ x

log p. (4.9)

Theorem 4.2.1. For every positive integer n ,

Π

p≤ n

p < 4n. (4.10)



Equivalently, for every real number x ≥ 1

ϑ (x ) < x log 4. (4.11)

Theorem 4.2.2. (Chebyshev)

There exist positive constants A and B such that

Ax ≤ ϑ (x ) ≤ ψ(x ) ≤ π(x ) log x ≤ Bx (4.12)

for all x ≥ 2. Moreover,

lim inf x →∞

ϑ (x )

x

= lim inf x →∞

ψ(x )

x = lim inf x →∞

π(x ) log x

x

≥ log 2 (4.13)

and

lim sup x →∞

ϑ (x )

x = lim sup x →∞

ψ(x )

x

= lim sup x →∞

π(x ) log x

x

≤ log 4. (4.14)4.3 The Prime Number TheoremTheorem 4.3.1. The Prime Num ber Theorem

Prime number theorem states that π(x ) is asymptotic to x/ log x , that is

lim x →∞

π(x ) log x

x

= 1. (4.15)

Chapter 5



Characters of A Finite Abelian GroupAnd Primes in Arithmetic

ProgressionsAs a certain part of the number theory, the discussion on the distribution

of primes is very

interesting. Here brings about how the homomorphisms(called characters)

from a finite abelian

group to multiplicative group of nonzero complex numbers play roles in the

distribution of

primes. Johann Peter Gustav Lejeune Dirichlet(1805-1859) can be mentioned

here as he

contributed a lot of ideas in the prime distribution. He developed many

theorems using the

notions of multiplicative characters or Dirichlet characters.

5.1 Characters of Finite Abelian GroupsDefinition 5.1.1. Let G be a finite abelian group. A group character is a

homomorphism

χ : G → C

×

, where C×

is the multiplicative group of nonzero complex numbers.The set of all characters of a finite abelian group G forms an abelian

group, called the dual

group or character group of G , and it is denoted by ˆ G.

Theorem 5.1.1. The dual of a cyclic group of order n is also a cyclic group of order n.Definition 5.1.2. Dir ichlet ch aracters

Let m be a positive integer and let G = (Z /m Z )× be the multiplicative group of units of

the

ring of integers modulo m . Corresponding to each character f of G, we define an

arithmetical

functionχ as follows:χ(n ) =

{

f (n + m Z ) if (n,m ) = 1;

0 if (n,m ) > 1.

The functionχ is called a Dirichlet character modulo m. The principal character χ0

is thecharacter which has the properties

χ(n ) =

{

1 if (n,m ) = 1;0 if (n,m ) > 1.



17

Theorem 5.1.2. There are ϕ(m ) distinct Dirichlet characters modulo m, each of

which is

completely multiplicative and periodic with period m .

i.e.,

χ(ab ) = χ(a )χ(b ) for all integers a and b

χ(a + m ) = χ(a ) for all integers a .

Conversely, if χ is completely multiplicative and periodic with period m, and if χ

(n )=0

whenever (n,m ) > 1, thenχ is one of the Dirichlet characters modulo m .

Proof. Let G = Z/m Z.

There are ϕ(m ) characters f of G , hence ϕ(m ) characters χ modulo m . The

multiplicative

property of χ follows from that of f . Again, the periodicity follows from

the fact that a ≡ b (mod m ) implies (a,m ) = (b,m ).

To prove the converse we note that the function f defined on the group G by

f (n + m Z) = χ(n )

if (n,m ) = 1, is a character of G and so a Dirichlet character modulo m .

Theorem 5.1.3. Orthogo nal i ty relat ions

Let Σ a(moΣ d m) denote the sum over a complete set of residue classes modulo m, and let

_(mod m) denote the sum over ϕ(m ) Dirichlet characters modulo m. If χ is a Dirichlet

character modulo m, thenΣ a (mod m)

χ(a ) =

{

ϕ(m ) if χ = χ0;

0 if χ = χ0.

If a is an integer, then

Σ _ (mod m)

χ(a ) ={

ϕ(m ) if a ≡ 1 (mod m );

0 if a ≡ 1 (mod m ).

5.2 Dirichlet L-FunctionsDefinition 5.2.1. Dir ichlet L-func t ion

Let χ be a Dirichlet character modulo m. The Dirichlet L-function associated withχ

is thefunction

L (s, χ) =

Σ∞

n=1



χ(n )

n s

where

s = σ + it

is a complex number with real part ℜ (s ) = σ and imaginary part ℑ (s ) = t .For example, if χ0 is the principal character modulo 3, then

L (s, χ0) = 1 +

1

2s +

1

4s +

1

5s +

1

7s +1

8s + . . . (5.1)

and if χ3 is the nonprincipal character modulo 3, then

L (s, χ3) = 1 − 1

2s +

1

4s

− 1

5s +

1

7s

− 1

8s + . . . (5.2)

Now, we put some basic theorems.

Theorem 5.2.1. Let χ be a Dirichlet character modulo m , and let s be a complex

number

with ℜ (s ) = σ > 1. The function L (s, χ) is analytic and has the Euler product

L (s, χ) =

Π p

(

1 − χ( p )

p s

)− 1

. (5.3)

Moreover, L (s, χ) = 0 and

log L (s, χ) =Σ



p

χ( p )

p s + O (1). (5.4)

Proof. Since ____

χ(n )

n s

____

≤ 1

n _

and Σ∞

n=1

1

n _

converges for σ > 1, it follows that the series L (s, χ) converges

uniformly and absolutely in

the half-plane σ ≥ 1 + δ for every δ > 0. Thus, the function L (s, χ) is

analytic. Similarly, for

every prime p , the series

Σ∞

k =0

χ( p k )

p ks

converges uniformly and absolutely in the half-plane σ > 1, and

Σ∞

k =0

χ( p k )

p ks =

(

1 − χ( p )

p s

)− 1

.

Since the character χ is completely multiplicative, the Fundamental

Theorem of Arithmetic

implies thatΠ p≤ x

( Σ∞

k =0

χ( p k )

p ks

)

=

Σ n∈ N ( x )

χ(n )



n s ,

where N denotes the set of all positive integers n divisible only by primes

p ≤ x . In particular,

if n ≤ x and p divides n , and so n ∈ N (x ).

For every ε > 0 there exists a number x 0 such that, if x ≥ x 0, then

Σ n>x

1

n _ < ε.

It follows that for x ≥ x 0, we have

_____

L (s, χ) −

Π

p≤ x

(

1 − χ( p )

p s

)− 1

_____

≤

Σ n>x

____

χ(n )n s

____

< ε.

Hence,

L (s, χ) =

Π p

(

1 − χ( p )

p s

)− 1

.

This product is called the Euler product of L (s, χ).

We shall show that L (s, χ) is nonzero for σ > 1. Since each factor of the

Euler product is

nonzero and so it suffices to prove that

Π p>x 0

(

1 − χ( p )



p s

)− 1

= 0.

In fact,

Π p>x 0

(

1 − χ( p )

p s

)− 1

> 0.

Now, for | z | < 1, the principal value of the logarithm has the power series

log

1

1 − z

=

Σ∞

1

z n

n

.

Applying this to the Dirichlet L -function for σ > 1, we get

log L (s, χ) = −

Σ p

log

(

1 − χ( p )

p s

)

(5.5)

=

Σ

pχ( p )

p s + O (1). (5.6)

Theorem 5.2.2. Let χ be a nonprincipal character modulo m . The Dirichlet L-

function

L (s, χ) is analytic in the half plane σ > 0. Let K be a compact set in the half-

planeσ > 0.

For s ∈ K and x ≥ 1,

L (s, χ) =

Σ

n≤ x

χ(n )



n s + O (x − _). (5.7)

where the implied constant depends on m and K .

5.3 Dirichlet’s Theorem for Primes of the Form 4n − 1

and 4n + 1Derichlet’s theorem states that if m ≥ 1 and a are relatively prime

integers, then the arithmetic

progression mk +a contains infinitely many primes p of the form p = mk +a .

One main theorem

is given below:

Theorem 5.3.1. For 1 < σ < 2,

Σ p≡ 1 (mod 4)

1

p _ =

1

2

log

1

σ − 1

+ O (1) (5.8)

and Σ p≡ 3 (mod 4)

1 p _ =

1

2

log

1

σ − 1

+ O (1). (5.9)

In particular, there exist infinitely many primes p ≡ 1(mod 4) and infinitely many

primes

p ≡ 3(mod 4).

In the above theorem, the limit of log 1

_− 1 tends to ∞ as σ tends to 1+.

Chapter 6Introduction to Additive Number

Theory



The aim of this study is to find the solution ofWaring’s problem. Waring’

s problem states that

for every integer k ≥ 2 there exists a positive integer h such that every

nonnegative integer

can be written as the sum of exactly h k th powers. Lagrange proved thatevery nonnegative

integer is a sum of four squares. This means that for every nonnegative

integer n there exist

nonnegative integers x 1, x 2, x 3, x 4 such that

n = x 2

1 + x 2

2 + x 2

3 + x 2

4 .

Similarly, Wieferich proved that every nonnegative integer is a sum of nine

cubes. Thus, for

every nonnegative integer n there exist nonnegative integers x 1, · · · , x 9

such that

n = x 3

1 + · · · + x 3

9.

In 1909, the German mathematician David Hilbert proved Waring’s problem

for all exponents

k . There is a natural generalization of Waring’s problem to polynomials.

We will discuss it

in the next chapter in details.

6.1 Sum of Four SquaresIn this section, we bring an independent proof of Waring’s probem when k =

2.

Lemma 6.1.1. (Euler) If the integers m and n are sum of four squares, then mn is

also asum of four squares.

Lemma 6.1.2. If p is an odd prime, then the congruencex 2 + y 2 + 1 ≡ 0 (mod p )

has a solution x 0, y 0, where 0 ≤ x 0 ≤ ( p − 1)/ 2 and 0 ≤ y 0≤ ( p − 1)/ 2. And, there

exists an

integer k < p such that kp is the sum of four squares.

22

Theorem 6.1.1. Any prime p can be written as the sum of four squares.

Theorem 6.1.2. (Langrange) Any nonnegative integer can be written as the sum of four

squares.



Chapter 7Waring’s Problem

7.1 Stable Bases• A set A of nonnegative integers is called a basis of order h if every

positive integer can

be written as the sum of exactly h elements of A. For example, by Lagrange

’s theorem

the set of squares is a basis of order four.

• Let A = { a i } ∞

i =0 be an infinite set of nonnegative integers such that a 0 < a 1 < a 2 < · · · .

The counting function of A, denoted by A(n ), counts the number of positive

elementsof A that do not exceed n , i.e.,

A(n ) =

Σ ai ∈ A

1≤ ai ≤ n

1. (7.1)

• The Shnirel’man density of the set A is

σ(A) = inf

{

A(n )

n

: n = 1, 2, ...

}

= sup

{

α :

A(n )

n

≥ α ∀ n = 1, 2, ...}

.

• Let B = { b i } ∞

i =0 be a set of nonnegative integers such that 0 = b 0 < b 1 < b 2 < · · · . We

construct the subset AB ⊆ A as follows:

AB = { a bi

} ∞

i =0.



Then a 0 = a b0< a b1< · · · . For example, AN0= A. If the Shnirel’man density of

B is

positive, then AB is called a subset of A of positive Shnirel’man density . And,

the set A

is called a stable basis if every subset of positive Shnirel’man density isa basis of finite

order.

• A set A is called an asymptotic basis of order h if every sufficiently large

positive integer

can be written as the sum of exactly h elements of A.

• The lower asymptotic density of a set A is

d L(A) = lim inf

{

A(n )n

: n = 1, 2, ...

}

.

Note: 0 ≤ d L(A) ≤ 1 for every set A.

24

7.2 Shnirel’man’s Theorem • Let A and B be two nonempty sets of integers. The sumset A + B is the set

consisting

of all integers of the form a + b , where a ∈ A and b ∈ B . Likewise, the

difference set

A − B can be defined.

Thus, |A + · {· z· + A}

h times

= hA.

And, A is a basis of order h if N0⊆ hA, i.e., if the sumset contains every

nonnegativeinteger.

Lemma 7.2.1. Let A and B be two sets of integers such that 0 ∈ A and 0 ∈ B . If A(n )

+

B (n ) ≥ n , then n ∈ A + B .

Proof. If n ∈ A, then n = n + 0 ∈ A + B . Similarly, if n ∈ B , then n = 0 + n

∈ A + B .

Suppose that n / ∈

A ∪ B . Define sets A′ and B ′ by



A′ = { n − a : a ∈ A, 1 ≤ a ≤ n − 1 } (7.2)

B ′ = B ∩ [1, n − 1]. (7.3)

Then | A′ | = A(n ), since n / ∈

A, and | B ′ | = B (n ), since n / ∈ B . Moreover,

A′ ∪ B ′ ⊆ [1, n − 1]. (7.4)

Also,

| A′ | + | B ′ | = A(n ) + B (n ) ≥ n, (7.5)

therefore we get A′ ∩ B ′ = ϕ. So, n − a = b for some a ∈ A and b ∈ B .

Lemma 7.2.2. Let A and B be two sets of integers such that 0 ∈ A and 0 ∈ B . If σ

(A) +

σ(B ) ≥ 1, then N0⊆ A + B .

Proof. We know that 0 = 0 + 0 ∈ A + B . For n ≥ 1, we have

A(n ) + B (n ) ≥ (σ(A) + σ(B ))n ≥ n. (7.6)

By using the above lemma, n ∈ A + B .

Lemma 7.2.3. Let A be a set of integers such that 0 ∈ A and σ(A) ≥ 1/ 2. Then A is

a basis

of order 2.

The proof of this lemma follows immediately by the previous lemma with A =

B .Theorem 7.2.1. (Shnirel’man) Let A and B be two sets of integers such that 0 ∈ A

and

0 ∈ B . Let σ(A) = α and σ(B ) = β. Then

σ(A + B ) ≥ α + β − αβ. (7.7)

Proof. Let n ≥ 1. Let a 0 = 0 and 1 ≤ a 1 < · · · < a k ≤ n be the k = A(n ) positive

elements

of A that do not exceed n . Since 0 ∈ B , it follows that a i = a i + 0 ∈ A + B

for i = 1, ..., k .For i = 1, ..., k − 1 let

1 ≤ a 1 < · · · < b r i

≤ a i +1 − a i − 1 (7.8)

be the r i = B (a i +1− a i − 1) positive integers in B that are less than a i +1− a i .

Then

a i < a i + b 1 < · · · < a i + b r i < a i +1 (7.9)

and

a i + b j ∈ A + B (7.10)



for j = 1, ..., r i . Let

1 ≤ b 1 < · · · ≤ n − a k (7.11)

be the r k = B (n − a k ) positive integers in B that do not exceed n − a k . Then

a k < a k + b 1 < · · · < a k + b r k

≤ n (7.12)

and

a k + b j ∈ A + B (7.13)

for j = 1, ..., r k . Thus it follows that

(A + B )(n ) ≥ A(n ) +

Σk

i =0

r i (7.14)

= A(n ) +

Σk − 1

i =0

B (a i +1− a i − 1) + B (n − a k ) (7.15)

≥ A(n ) + β

Σk − 1

i =0

(a i +1− a i − 1) + β(n − a k ) (7.16)

= (1 − β)A(n ) + βn (7.17)

≥ (α + β − αβ)n. (7.18)So,

(A + B )(n )

n

≥ α + β − αβ

for all positive integers n . Therefore,

σ(A + B ) = inf

{

(A + B )(n )

n

: n = 1, 2, ...

}

≥ α + β − αβ.

Theorem 7.2.2. Let h be a positive integer, and let A1, ...,Ah be sets of integers

with 0 ∈ Ai

for i = 1, ..., h . Then,

1 − σ(A1 + ... + Ah) ≤

Πh

i =1

(1 − σ(Ai )). (7.19)



Proof. We prove by induction on h . Let σ(Ai ) = αi for i = 1, ..., h . For h =

1, the result is

obvious. For h = 2, we got it in the above theorem.

Let h ≥ 3, and assume that the result is true for h − 1 sets. Let B = A1 + ·

· · + Ah− 1.Now,

1 − σ(A1 + · · · + Ah = 1 − σ(B + Ah) (7.20)

≤ (1 − σ(B ))(1 − σ(Ah)) (7.21)

≤

hΠ− 1

i =1

(1 − σ(Ai ))(1 − σ(Ah)) (7.22)

=

Πh

i =1

(1 − σ(Ai )). (7.23)

Theorem 7.2.3. Let 0 < α ≤ 1. There exists an integer h = h (α) such that if A1,

...,Ah are

sets of nonnegative integers with 0 ∈ Ai and σ(Ai ) ≥ α for all i = 1, ..., h , then

A1 + ... + Ah = N0. (7.24)

Proof. We have 0 ≤ 1 − α < 1, therefore, there exists a positive integer h 1

such that

0 ≤ (1 − α)h1≤ 1

2

.

Let h = 2h 1, and let A1+...+Ah be the sets of nonnegative integers with 0 ∈

Ai and σ(Ai ) ≥ α

for each i . By the above theorem,

σ(A) = σ(A1 + ... + Ah1) (7.25)

≥ 1 −

Πh1

i =1

(1 − σ(Ai )) (7.26)

≥ 1 − (1 − α)h1(7.27)

≥ 1

2

. (7.28)

Similarly,

σ(B ) = σ(Ah1+1 + ... + A2h1) ≥ 1

2



.

Thus,

A1 + ... + Ah = A + B = N0. (7.29)

Theorem 7.2.4. (Shnirel’man) Let A be a set of nonnegative integers such that 0 ∈

A and σ(A) > 0. Then A is a basis of finite order.

The proof follows from the above theorem with Ai = A.

7.3 Waring’s Problem for Polynomials The main aim of this section is to prove that for k ≥ 2, the set of

nonnegative k th powers is

a basis of finite order.

Let f (x ) be an integer-valued polynomial of degree k such that

A(f ) = { f (i ) } ∞

i =0

is a strictly increasing sequence of nonnegative integers. Without loss of

generality, we may

assume that gcd(A(f )) = 1.

Let NSE denote the the number of solutions of the equation. We define

representation

functions r f;s(n ) and R f;n(N ) for the polynomial f (x ) by

r f;s(n ) = NSE { f (x 1) + · · · + f (x s) = n : x 1, ..., x s ∈ N0 } (7.30)

and

R f;n(N ) =

Σ 0≤ n≤ N

r f;s(n ). (7.31)

Lemma 7.3.1. Let f (x ) =

Σk

i =0 a i be an integer-valued polynomial of degree k with leading

coefficient a k > 0. Let

x ∗ (f ) =

2(| a k − 1| + | a k − 2| + · · · + | a 0| )

a k

.

If x > x ∗ (f ) is an integer, then

a k x k

2

< f (x ) <

3a k x k

2

.

If N is sufficiently large, then

R f;s(N ) >



1

2

(

2N

3a k s

)s

k

.

Proof. The given polynomial can be written as

f (x ) = a k x k

(

1 +

a k − 1

a k x

+

a k − 2

a k x 2 + · · · +

a 0

a k x k

)

.

Now, for x > x ∗ (f )

____

f (x )

a k x k

− 1

____

=

____

a k − 1

a k x

+

a k − 2

a k x 2 + · · ·

+

a 0

a k x k

____

(7.32)

≤

| a k − 1|

a k x

+

| a k − 2|



a k x 2 + · · · +

| a 0|

a k x k (7.33)

≤

| a k − 1| + | a k − 2| + · · · + | a 0|

a k x

(7.34)

=

x ∗ (f )

2

(7.35)

<

1

2

. (7.36)

If x 1, ..., x s are integers such that

x ∗ (f ) < x j ≤

(

2N

3a k s

)1

k

for j − 1, ..., s , then0 <

a k x k

j

2

< f (x j ) <

3a k x k

j

2

≤ N

s and

0 < f (x 1) + · · · + f (x s) < N.

The number of integers in the interval

(

x ∗ (f ),

(

2N

3a k s

)1

k

]



is greater than (

2N

3a k s

)1

k

− x ∗ (f ) − 1,

and hence

R f;s(N ) >

((

2N

3a k s

)1

k

− x ∗ (f ) − 1

)s

≥ 1

2

(

2N

3a k s

)s

k

for N sufficiently large. This completes the proof.

Lemma 7.3.2. Let f (x ) =

Σk

i =0 a i be an integer-valued polynomial of degree k such that

A(f ) = { f (i ) } ∞ i =0

is a strictly increasing sequence of nonnegative integers. Let

N (f ) =

x ∗ (f )k

2k !

.

For N ≥ N (f ), if x 1, x 2, ..., x s are nonnegative integers with

Σs

j =1

f (x j ) ≤ N,

then

0 ≤ x j ≤ (2k !N )1k

for j = 1, ..., s .

The proof of this lemma follows from the fact that k !a k ≥ 1 and if N ≥ N (f )

and

x j > (2k !N )1=k ≥ x ∗ (f ), then

f (x j ) >



a k x k

j

2

≥ k !a k N ≥ N,

and

Σs

i =1

f (x i ) ≥ f (x j ) > N.

Theorem 7.3.1. Let { s (k ) } ∞

k =1 be the sequence of integers defined successively by s (1) = 1

and

s (k ) = 8k 2[log2 s(k − 1)]

for k ≥ 2. Let c ≥ 1 and P ≥ 1. If

f (x ) =

Σk

i =0

a i x i

is an integer-valued polynomial of degree k such that | a i | ≤ cP k − i for i = 0, 1, ...,

k , then for

every integer n ,

NSE

Σs(k )

j =1

f (x j ) = n with x j ∈ Z and | x j | ≤ cP forj = 1, ..., s (k )

≪ k;c P s(k )− k .

Theorem 7.3.2. Let f (x ) =

Σk

i =0 a i be an integer-valued polynomial of degree k with a k > 0

and gcd(A(f )) = 1. Then A(f ) ∪ { 0 } is an asymptotic basis of finite order, that is, for

someh and every sufficiently large integer n there exists a positive integer h n ≤ h and

nonnegative

integers x 1, ..., x hn such that

f (x 1) + · · · + f (x hn) = n.

Proof. Define N (f ) by

N (f ) =

x ∗ (f )k

2k !

as above. Let s = s (k ) be the integer constructed above. Let W = sA(f ) bethe set consisting



of all sums of s integers of the form f (x ) with x ∈ N0. We shall show that

the sumset W has

lower asymptotic density d L(W ) > 0.

Let W (N ) be the counting function of W . We select c ≥ (2k !)1=k and N > N (f )

sufficiently

large that for P = N 1=k ,

| a i | ≤ cP k − 1

for i = 0, 1, ..., k . Then 0 < a k Σ ≤ c . Now, if x 1, ..., x s are nonnegative

integers such that s

j =1 f (x j ) ≤ N , then for j = 1, ..., s ,

0 ≤ x j ≤ (2k !)1=k ≤ cP.

Now, we get upper bounds for r f;s(n ) and R f;s(N ) as follows: If 0 ≤ n ≤ N ,

then

r f;s(n ) = NSE { f (x 1) + · · · + f (x s) = n : x i ∈ N0 } (7.37)

≤ NSE { f (x 1) + · · · + f (x s) = n : | x j | ≤ cP } (7.38)

≪ k;c P s− k (7.39)

and

R f;s(N ) =

Σ 0≤ n≤ N

r f;s(n ) (7.40)=

Σ 0≤ n≤ N

r f;s(n)≥ 1

r f;s(n ) (7.41)

≪ k;c W (N )P s− k (7.42)

≪ k;c

(

W (N )

N

)

P s. (7.43)

Now, for sufficiently large N ,

R f;s(N ) >

1

2

(

2N

3a k s

)s

k



≥ 1

2

(

2N

3cs )s

k

≫ k;c P s.

Thus,

P s ≪ k;c R f;s(N ) ≪ k;c

(

W (N )

N

)

P s,

and so W (N ) ≫ k;c 1. Hence,

d L(sA(f )) = d L(W ) > 0.

Theorem 7.3.3. Let f (x ) be an integer-valued polynomial of degree k with leading

coefficient

a k > 0. If 0, 1 ∈ A(f ) = { f (x ) : x ∈ N0 } , then A(f ) is a basis of finite order.

The proof follows from the above theorem.

Theorem 7.3.4. (Waring-Hilbert) For every k ≥ 2, the set of nonnegative k th powers

is

a basis of finite order.Proof. The proof can be obtained from the above theorem by applying to the

polynomial

f (x ) = x k .

Chapter 8Sums of Sequences of Polynomials8.1 Sums and Differences of Weighted Sets

Definition 8.1.1. A weighted set is a pair (A,w A)

, whereA

is a set and w A

is afunction(called

the weight function) defined on A.

Note: In this chapter, weighted sets are always finite set of integers, and

the range of the

weight functions is the set of nonnegative integers, i.e., w A(a ) ∈ N0 for

all a ∈ A. Thus a

weighted set can be regarded as the set with multiplicities.

There are ways to get the weighted sets. Few important weighted sets are

discuss below:



• If (A,w A) is a weighted set and A is a subset of A∗ , then we can define

the weighted set

(A∗ ,w A∗ ) by

w A∗ (a ) =

{w A(a ) if a ∈ A

0 if a ∈ A∗ A.

• Let (A1,w A1), ..., (Ah,w Ah) be weighted sets. We define a weight function

on the product

set by

w A1×···× Ah(a 1, ..., a h) = w A1(a 1) · · ·w Ah(a h).

• Let f : A1,w A1), ..., (Ah,w Ah

→ B be a function defined on the product set. We define

a weight function w (f )

B on B as follows:

w (f )

B (b ) =

Σ (a1;:::;ah)∈ A1×···× Ah

f (a1;:::;ah)=b

w A1×···× Ah(a 1, ..., a h) (8.1)

=

Σ (a1;:::;ah)∈ A1×···× Ah

f (a1;:::;ah)=b

w A1(a 1) · · ·w Ah(a h). (8.2)

We can think of w (f )

B as counting the weighted number of solutions of the equation

f (a 1, ..., a h) = b

• Let f : A1,w A1), ..., (Ah,w Ah

→ B be a function defined on the product set. Then

NSE { f (a 1, ..., a h) = b } =

Σ (a1;:::;ah)∈ A1×···× Ah

f (a1;:::;ah)=b

1

with each a i ∈ Ai .

32

Theorem 8.1.1. For l ≥ 2, let h, r 0, r 1, ..., r l be the integers such that

0 = r 0 < r 1 < · · · < r l = h.



Let (A1,w A1), ..., (Ah,w Ah) be weighted sets and let B 1, ...,B l , and C be sets. For

i = 1, ..., l ,

let

f i : Ar i − 1+1× · · · × Ar i

→ B i

be a function defined on the weighted product set Ar i − 1+1×· · ·× Ar i . Then f i induces a

weight

function w (f i )

Bi on the set B I , and these weight functions determine a weight function on the

product set B 1× · · · × B l . Let

g : B 1 × · · · × B l → C

be a function defined on the weighted product set B 1 × · · · × B l . Then g induces a

weight function w (g )

C on C . Define the function

f : A1 × · · · × Ah→ C

by

f (a 1, ..., a h) = g (f 1(a 1, ..., a r 1), f 2(a r 1+1, ..., a r 2), ..., f 2(a r l − 1+1, ...,

a r l )).

Then f induces a weight function w (f )

C on C . For all c ∈ C we have

w (f )C (c ) = w (g )

C (c ),

that is,Σ (a1;:::;ah)∈ A1×···× Ah

f (a1;:::;ah)=c

w A1×···× Ah(a 1, ..., a h) =

Σ (b1;:::;bl )∈ B1×···× Bl

f (b1;:::;bl )=c

w B1×···× Bl (b 1, ..., b l ).Proof. The proof is simple.

w (g )

C (c ) =

Σ (b1;:::;bl )∈ B1×···× Bl

g (b1;:::;bl )=c

w B1×···× Bl (b 1, ..., b l ) (8.3)

=

Σ (b1;:::;bl )∈ B1×···× Bl

g (b1;:::;bl )=c

w (f 1)



B1(b 1) · · ·w (f l )Bl

(b l ) (8.4)

=

Σ (b1;:::;bl )∈ B1×···× Bl

g (b1;:::;bl )=c

Σ (a1;:::;ar 1 )∈ A1×···× Ar 1

f 1(a1;:::;ar 1)=b1

Πr 1

i =1

w Ai (a i )

× · · · (8.5)

×

Σ (arl − 1+1;:::;arl )∈ Arl − 1+1×···× Arl

f l (arl − 1+1;:::;arl )=bl

Πr l

i =r l − 1+1

w Ai (a i )

(8.6)

=

Σ (b1;:::;bl )∈ B1×···× Bl

g (b1;:::;bl )=c

Σ (a1;:::;ar 1 )∈ A1×···× Ar 1

f 1(a1;:::;ar 1)=b1

· · · (8.7)Σ (arl − 1+1;:::;arl )∈ Arl − 1+1×···× Arl

f l (arl − 1+1;:::;arl )=bl

Πh

i =1

w Ai (a i ) (8.8)

=

Σ (a1;:::;ah)∈ A1×···× Ah

g (f 1(a1;:::;ar 1 );:::;f 2(arl − 1+1;:::;arl ))=c

Πhi =1



w Ai (a i ) (8.9)

=

Σ (a1;:::;ah)∈ A1×···× Ah

f (a1;:::;ah)=c

w A1×···× Ah(a 1, ..., a h) (8.10)= w (f )

C (c ). (8.11)

Lemma 8.1.1. Let B 1 and B 2 be the weighted set of integers. Define the addition map

σ : B 1× B 2→ B 1+B 2 by σ(b 1, b 2) = b 1+b 2 and the difference mapsδi : B i × B i → B i − B i

by

δi (b i , b ′

i ) = b i − b ′

i for i = 1, 2. Consider the weighted sumset S = B 1 + B 2 and the difference

sets D i = B i − B i for i = 1, 2. Then for all integer n , we have

w ( _)

S (n ) ≤ 1

2

(

w ( _1)

D1(0) + w ( _2)

D2(0)

)

.Lemma 8.1.2. For t ≥ 1, let B 1, ...,B 2t be weighted set of integers and let S be the

weighted sumset

S = B 1 + · · · + B 2t

with weight function determined by the addition mapσB 1 × · · · × B 2t → S . For i = 1,

..., 2t ,

consider the weighted difference sets

D i = 2t − 1B i − 2t − 1B i = 2t − 1(B i − B i )

with weight functions defined by the maps

δi : B i × · · · × B i → D i

δi (b i;1, ..., b i;2t ) = (b i;1 + · · · b i;2t − 1) − (b i;2t − 1+1 + · · · b i;2t ).

Then for all integers n ,

w ( _)

S (n ) ≤ 1

2t

2t Σ i =1

w ( _i )



Di (0).

Let B be a weighted sumset S = 2t B weighted difference set D = 2t − 1B − 2t − 1B . Then

w ( _)

S (n ) ≤ w ( _)

D (0) for all integers n ∈ S .

8.2 Linear and Quadratic EquationsLemma 8.2.1. Let Q ≥ 1. Let u 1, ..., u k be relatively prime integers such that

U = max{| u 1| , ..., | u k |} ≤ Q.

For every integer n ,

NSE { u 1v 1 + · · · + u k v k = n } ≤ (k − 1)!(3Q )k − 1

U

.

This lemma simply says that we can define weighted sets Ai = { v ∈ Z : | v | ≤ Q }

with

weights 1 for all v and w (f )

B (n ) ≤ (k − 1)!(3Q)k − 1

U where f (v 1, ..., v k ) = u 1v 1 + · · · + u k v k .

Theorem 8.2.1. Let k ≥ 3 and let P,Q and c be real numbers such that

1 ≤ P ≤ Q ≤ cP k − 1.

Consider the quadratic equation

u 1v 1 + · · · + u k v k = 0.

Then

NSE { u 1v 1 + · · · + u k v k = 0 } ≪ k;c (PQ )k − 1

provided | u i | ≤ P and | v i | ≤ Q for i = 1, ..., k .

Bibliography[1] Melvyn B. Nathanson, Methods in Number Theory of Gratuate Text in

Mathematics,

Springer-Verlag, New York, Third Indian Reprint, 2009.

[2] Tom M. Apostol, Introduction To Analytic Number Theory , First Narosa

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House Reprint, 1989, Copyright c⃝

by Springer-Verlag, New York Inc.

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analytic methods in number theory

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