Skip to document
Welcome to StudocuSign in to access the best study resources
Guest user
0followers
HomeMy LibraryAsk AI
  • You don't have any recent items yet.
My Library
  • You don't have any modules yet.
  • You don't have any books yet.
  • You don't have any Studylists yet.

Number Theory 2017-2018 Example Sheet 3

Module

Number theory

5 Documents
Students shared 5 documents in this course
Academic year: 2017/2018
Uploaded by:
Anonymous Student
This document has been uploaded by a student, just like you, who decided to remain anonymous.
Newcastle University

Comments

Please sign in or register to post comments.

Students also viewed

Related documents

Preview text

Michaelmas Term 2017 Number Theory: Example Sheet 3 of 4

Throughout this sheet,φdenotes the Euler totient function,μthe M ̈obius function,τ(n) the number of positive divisors ofn, andσ(n) the sum of the positive divisors ofn.

  1. Prove that for Re(s)>1, we have

ζ(s) 2 =

∑∞

n=

τ(n) ns

.

Can you find Dirichlet series for 1/ζ(s) andζ(s−1)/ζ(s) (for suitable values ofs)?

  1. Find all natural numbersnfor whichσ(n) +φ(n) =nτ(n).

  2. (i) Define the M ̈obius functionμ, and check that it is multiplicative. (ii) Let∑ f be a function defined on the natural numbers, and definegbyg(n) = d|nμ(d)f(

n d). Find an expression forfin terms ofg. (iii) Find a relationship betweenμandφ.

  1. Compute

∑

d|nΛ(d) for natural numbersn. (Here Λ is the von Mangoldt function.) 5. Use Legendre’s formula to computeπ(207).

  1. LetN be a positive integer greater than 1.

(i) Show that the exact power of a primepdividingN! is

∑∞

k=1⌊

N pk⌋. (ii) Prove the inequalityN!>(Ne)N. (iii) Deduce that ∑

p 6 N

logp p− 1

>(logN)− 1.

  1. Prove that every non-constant polynomial with integer coefficients assumes in- finitely many composite values.

  2. Prove that every integerN >6 can be expressed as a sum of distinct primes. (One method is to find a strictly increasing sequence of integers (ak) such that every integer 6< N 6 akis a sum of distinct primes less than or equal to thekth prime.)

  3. Prove that for everyn>1, the set of numbers{ 1 , 2 ,... , 2 n}can be partitioned into pairs{a 1 , b 1 },{a 2 , b 2 },... ,{an, bn}so that the sumai+biof each pair is prime.

  4. Calculatea 0 ,... , a 4 in the continued fraction expansions ofeandπ.

  5. Letabe a positive integer. Determine explicitly the real numberwhose continued fraction is [a, a, a,.. .].

  6. Determine the continued fraction expansions of

√

3,

√

7,

√

13,

√

19.

a.j@dpmms.cam.ac - 1 - 13 November 2017

  1. Letdbe a positive integer that is not a square. Letθnandpn/qnbe the complete quotients and convergents arising in the continued fraction expansion of

√

d. Show that for alln>1 we havepn− 1 −qn− 1

√

d= (−1)n/

∏n i=1θi. 14. (Extra question, requires Analysis II.) Letχ 4 be the non-trivial group homomor- phism (Z/ 4 Z)×→{± 1 }. Show that

L(s, χ 4 ) = 1−

1

3 s

+

1

5 s

−

1

7 s

+

1

9 s

−

1

11 s

+...

is a continuous function on (0,∞) withL(1, χ 4 ) 6 = 0. Use the Euler products to show that fors >1 we have

logζ(s) =

∑

p

1

ps

+g 1 (s)

logL(s, χ 4 ) =

∑

p 6 =

χ 4 (p) ps

+g 2 (s)

whereg 1 andg 2 are bounded functions. Deduce a special case of Dirichlet’stheorem on primes in arithmetic progression.

a.j@dpmms.cam.ac - 2 - 13 November 2017

Was this document helpful?

Number Theory 2017-2018 Example Sheet 3

Module: Number theory

5 Documents
Students shared 5 documents in this course

University: University of Cambridge

Was this document helpful?
Michaelmas Term 2017
Number Theory: Example Sheet 3 of 4
Throughout this sheet, φdenotes the Euler totient function, µthe M¨obius function, τ(n)
the number of positive divisors of n, and σ(n) the sum of the positive divisors of n.
1. Prove that for Re(s)>1, we have
ζ(s)2=
∞
X
n=1
τ(n)
ns.
Can you find Dirichlet series for 1/ζ(s) and ζ(s−1)/ζ(s) (for suitable values of s)?
2. Find all natural numbers nfor which σ(n) + φ(n) = nτ(n).
3. (i) Define the M¨obius function µ, and check that it is multiplicative.
(ii) Let fbe a function defined on the natural numbers, and define gby g(n) =
Pd|nµ(d)f(n
d). Find an expression for fin terms of g.
(iii) Find a relationship between µand φ.
4. Compute Pd|nΛ(d) for natural numbers n. (Here Λ is the von Mangoldt function.)
5. Use Legendre’s formula to compute π(207).
6. Let Nbe a positive integer greater than 1.
(i) Show that the exact power of a prime pdividing N! is P∞
k=1⌊N
pk⌋.
(ii) Prove the inequality N!>(N
e)N.
(iii) Deduce that
X
p6N
log p
p−1>(log N)−1.
7. Prove that every non-constant polynomial with integer coefficients assumes in-
finitely many composite values.
8. Prove that every integer N > 6 can be expressed as a sum of distinct primes. (One
method is to find a strictly increasing sequence of integers (ak) such that every
integer 6 < N 6akis a sum of distinct primes less than or equal to the kth prime.)
9. Prove that for every n>1, the set of numbers {1,2,...,2n}can be partitioned
into pairs {a1, b1},{a2, b2},...,{an, bn}so that the sum ai+biof each pair is prime.
10. Calculate a0, . . . , a4in the continued fraction expansions of eand π.
11. Let abe a positive integer. Determine explicitly the real number whose continued
fraction is [a, a, a, . . .].
12. Determine the continued fraction expansions of √3, √7, √13, √19.
a.j.scholl@dpmms.cam.ac.uk - 1 - 13 November 2017

Students also viewed

Related documents