aarjaneiro/planetmath huggingface dataset:
quadratic reciprocity rule

quadratic reciprocity rule

Theorem

(Law of Quadratic Reciprocity).

Let

p

and

q

be two distinct

odd primes

. Then:

(

q

p

)

⁢

(

p

q

)

=

(

-

1

)

(

p

-

1

)

⁢

(

q

-

1

)

/

4

where

(

⋅

⋅

)

is the Jacobi (

http://planetmath.org/JacobiSymbol

) symbol (or

Legendre symbol

).

The following is an

equivalent

formulation of the Law of Quadratic Reciprocity:

Theorem

(Quadratic Reciprocity (second form)).

Let

p

,

q

be distinct odd primes. Then:

1.

(

p

q

)

=

(

q

p

)

if one of

p

or

q

is

congruent

to

1

modulo

4

;

2.

(

p

q

)

=

-

(

q

p

)

if both

p

and

q

are congruent to

3

modulo

4

.

Title

quadratic reciprocity rule

Canonical name

QuadraticReciprocityRule

Date of creation

2013-03-22 11:42:27

Last modified on

2013-03-22 11:42:27

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

33

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11A15

Synonym

quadratic reciprocity

Related topic

EulersCriterion

Related topic

CubicReciprocityLaw

Related topic

QuadraticReciprocityForPolynomials

Related topic

LegendreSymbol

Generated on Fri Feb 9 15:12:04 2018 by

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quadratic residue

quadratic residue

Let

a

,

n

be relatively

prime integers

. If there exists an

integer

x

that satisfies

x

2

≡

a

(

mod

n

)

then

a

is said to be a

quadratic residue

of

n

. Otherwise,

a

is called a

quadratic nonresidue

of

n

.

Title

quadratic residue

Canonical name

QuadraticResidue

Date of creation

2013-03-22 11:55:19

Last modified on

2013-03-22 11:55:19

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11A15

Related topic

LegendreSymbol

Related topic

EulersCriterion

Defines

quadratic non-residue

Defines

quadratic nonresidue

Generated on Fri Feb 9 15:12:06 2018 by

LaTeXML
representants of quadratic residues

representants of quadratic residues

Theorem.

Let

p

be a

positive

odd prime

number.  Then the

integers

1

2

,

2

2

,

…

,

(

p

-

1

2

)

2

(1)

constitute a representant system of incongruent

quadratic residues

modulo

p

.  Accordingly, there are

p

-

1

2

quadratic residues and equally many nonresidues modulo

p

.

Proof.

Firstly, the numbers (1), being squares, are quadratic residues modulo

p

.  Secondly, they are incongruent, because a

congruence

a

2

≡

b

2

(

mod

p

)

would imply

p

∣

a

+

b

or

p

∣

a

-

b

,

which is impossible when

a

and

b

are different integers among

1

,

2

,

…

,

p

-

1

2

.  Third, if

c

is any quadratic residue modulo

p

, and therefore the congruence

x

2

≡

c

(

mod

p

)

has a solution

x

, then

x

is

congruent

with one of the numbers

±

1

,

±

2

,

…

,

±

p

-

1

2

which form a

reduced residue system

modulo

p

(see absolutely least remainders).  Then

x

2

and

c

are congruent with one of the numbers (1).

Title

representants of quadratic residues

Canonical name

RepresentantsOfQuadraticResidues

Date of creation

2013-03-22 19:00:35

Last modified on

2013-03-22 19:00:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A15

Synonym

representant system of quadratic residues

Related topic

GaussianSum

Related topic

DifferenceOfSquares

Related topic

DivisibilityByPrimeNumber

Generated on Fri Feb 9 15:12:08 2018 by

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2^{ω⁢(n)}≤τ⁢(n)≤2^{Ω⁢(n)}

2

ω

⁢

(

n

)

≤

τ

⁢

(

n

)

≤

2

Ω

⁢

(

n

)

Throughout this entry,

ω

,

τ

, and

Ω

denote the

number of distinct prime factors function

, the

divisor function

, and the number of (nondistinct)

prime factors

function

(

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

), respectively.

Theorem

.

For any

positive integer

n

,

2

ω

⁢

(

n

)

≤

τ

⁢

(

n

)

≤

2

Ω

⁢

(

n

)

.

Proof.

Note that

2

ω

⁢

(

n

)

,

τ

⁢

(

n

)

, and

2

Ω

⁢

(

n

)

are

multiplicative

. Also note that, for any positive integer

n

, the numbers

2

ω

⁢

(

n

)

,

τ

⁢

(

n

)

, and

2

Ω

⁢

(

n

)

are positive integers. Therefore, it will suffice to prove the inequality for prime powers.

Let

p

be a prime and

k

be a positive integer. Thus:

2

ω

⁢

(

p

k

)

=

2

τ

⁢

(

p

k

)

=

k

+

1

2

Ω

⁢

(

p

k

)

=

2

k

Hence,

2

ω

⁢

(

p

k

)

≤

τ

⁢

(

p

k

)

≤

2

Ω

⁢

(

p

k

)

. It follows that

2

ω

⁢

(

n

)

≤

τ

⁢

(

n

)

≤

2

Ω

⁢

(

n

)

.
∎

This theorem has an obvious corollary.

Corollary

.

For any

squarefree

positive integer

n

,

2

ω

⁢

(

n

)

=

τ

⁢

(

n

)

=

2

Ω

⁢

(

n

)

.

Title

2

ω

⁢

(

n

)

≤

τ

⁢

(

n

)

≤

2

Ω

⁢

(

n

)

Canonical name

2omeganletaunle2Omegan

Date of creation

2013-03-22 16:07:28

Last modified on

2013-03-22 16:07:28

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

14

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11A25

Related topic

NumberOfDistinctPrimeFactorsFunction

Related topic

TauFunction

Related topic

NumberOfNondistinctPrimeFactorsFunction

Related topic

DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2

Generated on Fri Feb 9 15:12:14 2018 by

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additive function

additive function

In

number theory

, an

additive function

is an

arithmetic function

f

:

ℕ

→

ℂ

with the property that

f

⁢

(

1

)

=

0

and, for all

a

,

b

∈

ℕ

with

gcd

⁡

(

a

,

b

)

=

1

,

f

⁢

(

a

⁢

b

)

=

f

⁢

(

a

)

+

f

⁢

(

b

)

.

An arithmetic function

f

is said to be

completely additive

if

f

⁢

(

1

)

=

0

and

f

⁢

(

a

⁢

b

)

=

f

⁢

(

a

)

+

f

⁢

(

b

)

holds for

all

positive integers

a

and

b

, when they are not relatively prime. In this case, the

function

is a

homomorphism

of monoids and, because of the

fundamental theorem of arithmetic

, is completely determined by its

restriction

to

prime numbers

. Every completely additive function is

additive

.

Outside of number theory, the additive is usually used for all functions with the property

f

⁢

(

a

+

b

)

=

f

⁢

(

a

)

+

f

⁢

(

b

)

for all

arguments

a

and

b

. (For

instance

, see the other entry titled additive function (

http://planetmath.org/AdditiveFunction2

).) This entry discusses number theoretic additive functions.

Additive functions cannot have

convolution inverses

since an arithmetic function

f

has a convolution inverse if and only if

f

⁢

(

1

)

≠

0

. A proof of this equivalence is supplied here (

http://planetmath.org/ConvolutionInversesForArithmeticFunctions

).

The most common of additive function in all of mathematics is the

logarithm

. Other additive functions that are useful in number theory are:

•

ω

⁢

(

n

)

, the

number of distinct prime factors function

•

Ω

⁢

(

n

)

, the number of (nondistinct)

prime factors

function (

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

)

By exponentiating an additive function, a

multiplicative function

is obtained. For example, the function

2

ω

⁢

(

n

)

is multiplicative. Similarly, by exponentiating a completely additive function, a

completely multiplicative function

is obtained. For example, the function

2

Ω

⁢

(

n

)

is completely multiplicative.

Title

additive function

Canonical name

AdditiveFunction1

Date of creation

2013-03-22 16:07:03

Last modified on

2013-03-22 16:07:03

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

12

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11A25

Related topic

MultiplicativeFunction

Defines

additive

Defines

completely additive

Defines

completely additive function

Generated on Fri Feb 9 15:12:16 2018 by

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aliquot sequence

aliquot sequence

For a given

m

, define the

recurrence relation

a

1

=

m

,

a

n

=

σ

⁢

(

a

n

-

1

)

-

a

n

-

1

, where

σ

⁢

(

x

)

is the

sum of divisors function

.

a

is then the

aliquot sequence

of

m

.

If

m

is an

amicable number

, its aliquot sequence is

periodic

, alternating between the abundant and deficient member of the amicable pair. For a

prime number

p

, its aliquot sequence is

p

,

1

,

0

. In other cases, the aliquot sequence reaches a

fixed point

upon 0, or on a

perfect number

.

Title

aliquot sequence

Canonical name

AliquotSequence

Date of creation

2013-03-22 16:07:14

Last modified on

2013-03-22 16:07:14

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Generated on Fri Feb 9 15:12:18 2018 by

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alternate proof of Möbius inversion formula

alternate proof of Möbius inversion formula

The Möbius

inversion

theorem

can also be proved elegantly using the fact that

arithmetic functions

form a ring under

+

and

*

.

Let

I

be the arithmetic function that is everywhere

1

. Then obviously if

μ

is the Möbius

function

,

(

μ

*

I

)

⁢

(

n

)

=

∑

d

|

n

μ

⁢

(

d

)

⁢

I

⁢

(

n

d

)

=

∑

d

|

n

μ

⁢

(

d

)

=

{

1

n=1

0

otherwise

and thus

I

*

μ

=

e

, where

e

is the

identity

of the ring.

But then

f

⁢

(

n

)

=

∑

d

|

n

g

⁢

(

d

)

=

∑

d

|

n

g

⁢

(

d

)

⁢

I

⁢

(

n

d

)

and so

f

=

g

*

I

. Thus

f

*

μ

=

g

*

I

*

μ

=

g

. But

g

=

f

*

μ

means precisely that

g

⁢

(

n

)

=

∑

d

|

n

μ

⁢

(

d

)

⁢

f

⁢

(

n

d

)

and we are done.

The reverse equivalence is

similar

(

f

*

μ

=

g

⇒

f

*

μ

*

I

=

g

*

I

⇒

f

=

g

*

I

).

Title

alternate proof of Möbius inversion

formula

Canonical name

AlternateProofOfMobiusInversionFormula

Date of creation

2013-03-22 16:30:31

Last modified on

2013-03-22 16:30:31

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Proof

Classification

msc 11A25

Generated on Fri Feb 9 15:12:23 2018 by

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another proof that a number is polite iff it is positive and not a positive power of 2

another proof that a number is polite iff it is positive and not a positive power of

2

In this entry we give another proof that an

integer

is polite iff it is neither non-positive nor a

positive

power of

2

. The proof utilizes the

formula

a

+

(

a

+

1

)

+

⋯

+

b

=

(

a

+

b

)

⁢

(

b

-

a

+

1

)

2

.

Proof.

By definition, an integer

n

is polite if it a sum of consecutive non-negative integers,

n

itself must be non-negative. Furthermore

n

can not be

0

since a sum of at least two consecutive non-negative integers must be positive. So we may assume that

n

is positive.

There are two cases:

1.

n

is a power of

2

:

Suppose that

n

is polite, say

n

=

a

+

(

a

+

1

)

+

⋯

+

(

a

+

k

)

, where

a

is non-negative and

k

>

0

, then

n

=

(

2

⁢

a

+

k

)

⁢

(

k

+

1

)

2

This means that

(

2

⁢

a

+

k

)

⁢

(

k

+

1

)

=

2

⁢

n

is a power of

2

, or

2

⁢

a

+

k

and

k

+

1

are both powers of

2

by the

unique factorization

of

positive integers

. Since

k

>

0

,

k

+

1

>

1

, so that if

k

+

1

were a power of

2

,

k

must be odd, which implies that

2

⁢

a

+

k

is odd too. Since

2

⁢

a

+

k

is a power of

2

, this forces

2

⁢

a

+

k

=

1

. As

k

>

0

and

a

≥

0

, there is only one solution:

k

=

1

and

a

=

0

, or

n

=

1

, showing that

1

is the only power of

2

that is polite.

2.

n

is not a power of

2

:

Let

p

be the smallest

odd prime

dividing

n

. Write

n

=

m

⁢

p

. So

m

≥

p

, or

m

-

p

≥

0

. Set

a

:=

2

⁢

m

+

1

-

p

2

.

Since

2

⁢

m

+

1

-

p

is the sum of

2

⁢

m

and

1

-

p

, both

even numbers

,

a

is an integer. Since

2

⁢

m

+

1

-

p

=

(

m

-

p

)

+

(

m

+

1

)

≥

m

+

1

>

0

,

a

is positive. Solving for

m

we get

m

=

2

⁢

a

+

p

-

1

2

.

Then

a

+

(

a

+

1

)

+

⋯

+

(

a

+

p

-

1

)

=

(

2

⁢

a

+

p

-

1

)

⁢

p

2

=

m

⁢

p

=

n

,

showing that

n

is polite.

∎

Title

another proof that a number is polite iff it is positive and not a positive power of

2

Canonical name

AnotherProofThatANumberIsPoliteIffItIsPositiveAndNotAPositivePowerOf2

Date of creation

2013-03-22 18:10:05

Last modified on

2013-03-22 18:10:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 11A25

Generated on Fri Feb 9 15:12:32 2018 by

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applications of second order recurrence relation formula

applications of second order recurrence relation formula

We give two applications of the

formula for sequences satisfying second order recurrence relations

:

1.

Recall that the

Fibonacci sequence

satisfies

the

recurrence relation

f

n

+

1

=

f

n

+

f

n

-

1

.

Thus,

f

0

=

1

,

A

=

1

, and

B

=

1

. Therefore, the

theorem

yields the following

formula

for the Fibonacci sequence:

f

n

=

∑

k

=

0

⌊

n

2

⌋

(

n

-

k

k

)

2.

Fix (

http://planetmath.org/Fix2

) a prime

p

and define a

sequence

s

by

s

n

=

τ

⁢

(

p

n

)

, where

τ

denotes the

Ramanujan tau function

. Recall that

τ

satisfies

τ

⁢

(

p

n

+

1

)

=

τ

⁢

(

p

)

⁢

τ

⁢

(

p

n

)

-

p

11

⁢

τ

⁢

(

p

n

-

1

)

.

Thus,

s

0

=

1

,

A

=

τ

⁢

(

p

)

, and

B

=

-

p

11

. Therefore, the theorem yields

τ

⁢

(

p

n

)

=

∑

k

=

0

⌊

n

2

⌋

(

n

-

k

k

)

⁢

(

-

p

11

)

k

⁢

(

τ

⁢

(

p

)

)

n

-

2

⁢

k

.

This formula is valid for all primes

p

and all

nonnegative integers

n

.

Title

applications of second order recurrence relation formula

Canonical name

ApplicationsOfSecondOrderRecurrenceRelationFormula

Date of creation

2013-03-22 17:51:46

Last modified on

2013-03-22 17:51:46

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Application

Classification

msc 11A25

Classification

msc 11F11

Classification

msc 11B39

Classification

msc 11B37

Classification

msc 03D20

Related topic

FibonacciSequence

Related topic

RamanujanTauFunction

Generated on Fri Feb 9 15:12:35 2018 by

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arithmetic function

arithmetic function

An

arithmetic function

is a

function

f

:

ℤ

+

→

ℂ

from the

positive integers

to the

complex numbers

.

Any

algebraic function

over

ℤ

+

, as well as

transcendental functions

such as

sin

⁡

(

n

⁢

π

)

and

e

n

⁢

π

⁢

i

with

n

∈

ℤ

+

are arithmetic functions.

There are two noteworthy operations on the set of arithmetic functions:

If

f

and

g

are two arithmetic functions, the

sum

of

f

and

g

, denoted

f

+

g

, is given by

(

f

+

g

)

⁢

(

n

)

=

f

⁢

(

n

)

+

g

⁢

(

n

)

,

and the

Dirichlet convolution

of

f

and

g

, denoted by

f

*

g

, is given by

(

f

*

g

)

⁢

(

n

)

=

∑

d

|

n

f

⁢

(

d

)

⁢

g

⁢

(

n

d

)

.

The set of arithmetic functions, equipped with these two binary operations, forms a commutative ring with unity. The 0 of the ring is the function

f

such that

f

⁢

(

n

)

=

0

for any positive integer

n

. The 1 of the ring is the function

f

with

f

⁢

(

1

)

=

1

and

f

⁢

(

n

)

=

0

for any

n

>

1

, and the units of the ring are those arithmetic function

f

such that

f

⁢

(

1

)

≠

0

.

Note that giving a

sequence

{

a

n

}

of complex numbers is equivalent to giving an arithmetic function by associating

a

n

with

f

⁢

(

n

)

.

Title

arithmetic function

Canonical name

ArithmeticFunction

Date of creation

2013-03-22 13:50:49

Last modified on

2013-03-22 13:50:49

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11A25

Related topic

ConvolutionInversesForArithmeticFunctions

Related topic

PropertyOfCompletelyMultiplicativeFunctions

Related topic

DivisorSumOfAnArithmeticFunction

Defines

Dirichlet convolution

Generated on Fri Feb 9 15:12:37 2018 by

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arithmetic functions form a ring

arithmetic functions form a ring

Theorem 1

The set

S

of

arithmetic functions

forms a

commutative ring

with unity under the operations of element-by-element

addition

and Dirichlet convolution, i.e. under

(

f

+

g

)

⁢

(

n

)

=

f

⁢

(

n

)

+

g

⁢

(

n

)

(

f

*

g

)

⁢

(

n

)

=

∑

d

|

n

f

⁢

(

d

)

⁢

g

⁢

(

n

d

)

The

0

of the ring is the function

z

such that

z

⁢

(

n

)

=

0

for all

positive integers

n

, the

1

of the ring is the

convolution identity function

ε

, and the units of the ring are those arithmetic functions

f

such that

f

⁢

(

1

)

≠

0

.

Proof.

This is essentially a triviality and a little bit of computation.

That

𝒮

is an

abelian group

under

+

is obvious; the only interesting is noting that indeed

z

is the

identity

of the group (the

0

of the ring).

Many of the ring identities are also obvious. We will prove that

ε

is the

multiplicative identity

, that

*

is

commutative

and

associative

, that

*

distributes over

+

, and that the units of the ring are as stated.

To see that

ε

is the multiplicative identity, note that

(

ε

*

f

)

⁢

(

n

)

=

∑

d

|

n

ε

⁢

(

d

)

⁢

f

⁢

(

n

d

)

=

ε

⁢

(

1

)

⁢

f

⁢

(

n

)

=

f

⁢

(

n

)

and thus

ε

*

f

=

f

.

To see that

*

is commutative, note that

f

*

g

can also be written as

(

f

*

g

)

⁢

(

n

)

=

∑

a

⁢

b

=

n

f

⁢

(

a

)

⁢

g

⁢

(

b

)

Commutativity is obvious from this of the operation.

Associativity follows similarly. Note that

(

(

f

*

g

)

*

h

)

⁢

(

n

)

=

∑

r

⁢

a

=

n

(

f

*

g

)

⁢

(

r

)

⁢

h

⁢

(

a

)

=

∑

r

⁢

a

=

n

h

⁢

(

a

)

⁢

∑

b

⁢

c

=

r

f

⁢

(

b

)

⁢

g

⁢

(

c

)

=

∑

a

⁢

b

⁢

c

=

n

f

⁢

(

b

)

⁢

g

⁢

(

c

)

⁢

h

⁢

(

a

)

If one expands

(

f

*

(

g

*

h

)

)

⁢

(

n

)

similarly, the resulting sum is identical, so the two are equal.

Distributivity follows since

(

f

*

(

g

+

h

)

)

⁢

(

n

)

=

∑

d

|

n

f

⁢

(

d

)

⁢

(

g

+

h

)

⁢

(

n

d

)

=

∑

d

|

n

f

⁢

(

d

)

⁢

(

g

⁢

(

n

d

)

+

h

⁢

(

n

d

)

)

=

∑

d

|

n

f

⁢

(

d

)

⁢

g

⁢

(

n

d

)

+

∑

d

|

n

f

⁢

(

d

)

⁢

h

⁢

(

n

d

)

=

(

(

f

*

g

)

+

(

f

*

h

)

)

⁢

(

n

)

The units of the ring are simply the invertible functions; the entry on

convolution inverses for arithmetic functions

shows that the invertible functions are those functions

f

with

f

⁢

(

1

)

≠

0

.

Title

arithmetic functions form a ring

Canonical name

ArithmeticFunctionsFormARing

Date of creation

2013-03-22 16:30:28

Last modified on

2013-03-22 16:30:28

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11A25

Related topic

ConvolutionInversesForArithmeticFunctions

Generated on Fri Feb 9 15:12:39 2018 by

LaTeXML
Basel problem

Basel problem

The

Basel problem

, first posed by Pietro Mengoli in 1644, asks for a finite formula for the infinite sum

∑

i

=

1

∞

1

i

2

.

Though Mengoli verified the Wallis formulae for

π

, it did not occur to him that

π

was also involved in the solution of this problem. Jakob Bernoulli also tried in vain to solve this problem. Even an approximate decimal value eluded contemporary mathematicians: an answer accurate to just five

decimal places

requires iterating up to at least

i

=

112000

, which without the aid of a

computer

was wholly impractical in Mengoli’s day. The problem was finally solved in 1741, when, after almost a decade of work, Leonhard Euler conclusively proved that

π

2

6

=

ζ

⁢

(

2

)

=

∑

i

=

1

∞

1

i

2

.

The value, 1.6449340668482264365… could then be computed to almost as many decimal places as were known of

π

. See value of the

Riemann zeta function

at

s

=

2

(

http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2

)

References

1

Ed Sandifer, “Euler’s Solution of the Basel Problem - The Longer Story”. Danbury, Connecticut: Western Connecticut State University (2003)

Title

Basel problem

Canonical

name

BaselProblem

Date of creation

2013-03-22 18:05:22

Last modified on

2013-03-22 18:05:22

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Generated on Fri Feb 9 15:12:41 2018 by

LaTeXML
Basel problem derivation

Basel problem derivation

The basis for this

derivation

is the

assumption

that the properties of finite

polynomials

hold true for infinite series. First, consider the Taylor series expansion of the sine function:

sin

⁡

x

=

∑

n

=

0

∞

(

-

1

)

n

(

2

⁢

n

+

1

)

!

⁢

x

2

⁢

n

+

1

=

x

-

x

3

3

!

+

x

5

5

!

-

x

7

7

!

+

⋯

Dividing through by x, we get:

sin

⁡

x

x

=

1

-

x

2

3

!

+

x

4

5

!

-

x

6

7

!

+

⋯

The roots of the

function

sin

⁡

x

/

x

occur at

x

=

n

⁢

π

, where

n

=

±

1

,

±

2

,

±

3

,

…

.
Let us assume that we can express this infinite series as a

product

of linear factors given by its roots, just as we do for finite polynomials:

sin

⁡

x

x

=

(

1

-

x

π

)

⁢

(

1

+

x

π

)

⁢

(

1

-

x

2

⁢

π

)

⁢

(

1

+

x

2

⁢

π

)

⁢

(

1

-

x

3

⁢

π

)

⁢

(

1

+

x

3

⁢

π

)

⁢

⋯

=

(

1

-

x

2

π

2

)

⁢

(

1

-

x

2

4

⁢

π

2

)

⁢

(

1

-

x

2

9

⁢

π

2

)

⁢

⋯

.

If we multiply out this product and collect all the

x

2

terms, we get:

-

(

1

π

2

+

1

4

⁢

π

2

+

1

9

⁢

π

2

+

⋯

)

=

-

1

π

2

⁢

∑

n

=

1

∞

1

n

2

From the original infinite series of

sin

⁡

x

/

x

the coefficient of

x

2

is

-

1

/

3

!

. Equating both expressions gives:

-

1

6

=

-

1

π

2

⁢

∑

n

=

1

∞

1

n

2

∑

n

=

1

∞

1

n

2

=

π

2

6

This was the method Euler used to solve the

Basel problem

. There are more

rigorous

and modern derivations, but this one is a simple, straight forward, and easily understood one.

Title

Basel problem derivation

Canonical name

BaselProblemDerivation

Date of creation

2013-03-22 18:40:50

Last modified on

2013-03-22 18:40:50

Owner

curious (18562)

Last modified by

curious (18562)

Numerical id

5

Author

curious (18562)

Entry type

Derivation

Classification

msc 11A25

Generated on Fri Feb 9 15:12:44 2018 by

LaTeXML