aarjaneiro/planetmath huggingface dataset:
square root of 3

square root of 3

The

square root of 3

, also known as

Theodorus’s constant

, is the number the square of which is equal to the

integer

3. It is an

irrational number

, one of the first few to have been proved irrational. Theodorus of Cyrene proved that the

square roots

of the integers 3, 5 to 8, 10 to 15 and 17 are all irrational. The

decimal expansion

of

3

is 1.7320508075688772935… (

sequence

http://www.research.att.com/ njas/sequences/A002194

A002194 in Sloane’s OEIS). Its

simple continued fraction

is

1

+

1

1

+

1

2

+

1

1

+

1

2

+

…

,

repeating 1 and 2 periodically (Sloane’s

http://www.research.att.com/ njas/sequences/A040001

A040001).

Given a unit cube, the diagonal from the vertex joining three sides to the other vertex joining the three other sides is

3

. Given a unit hexagon, the

distance

from one side to the parallel opposite side is

3

. More generally, the ratio of the length of a side of a hexagon to the distance from that side to the opposing parallel side is

1

:

3

, and the same ratio applies to the length of the side of a cube to the diagonal of that cube.

References

1

M. F. Jones, “22900D approximations to the square roots of the primes less than 100”,

Math. Comp

22

(1968): 234 - 235.

2

H. S. Uhler, “Approximations exceeding 1300 decimals for

3

,

1

3

,

sin

⁡

(

π

3

)

and

distribution

of digits in them”

Proc. Nat. Acad. Sci. U. S. A.

37

(1951): 443 - 447.

Title

square root of 3

Canonical name

SquareRootOf3

Date of creation

2013-03-22 17:29:46

Last modified on

2013-03-22 17:29:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Synonym

Theodorus’s constant

Synonym

Theodorus’ constant

Generated on Fri Feb 9 15:14:55 2018 by

LaTeXML
square root of 5

square root of 5

The

square root of 5

is an

irrational number

involved in

the formula for the

golden ratio

. It is also used in statistics when dealing with 5-business day weeks. Its

decimal expansion

begins 2.2360679774997896964, see

http://www.research.att.com/ njas/

sequences

/A002163

sequence A002163 in Sloane’s OEIS. Its

simple continued fraction

is 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, … One formula for the square root of 5 involves some of the same numbers as in Euler’s

identity

(but with a 2 instead of the 1):

e

i

⁢

π

+

2

⁢

ϕ

. The square root of 5 modulo a

prime number

is employed in some ECM algorithms.

A rectangle with unit height and

5

width can be split into two

golden rectangles

of the same size and a square, or into two golden rectangles of different sizes.

The conjecture stating “that any abelian surface with RM by

Q

⁢

(

5

)

is

isogenous

to a simple factor of the Jacobian of a modular curve

X

0

⁢

(

N

)

for some

N

” still stands. John Wilson has produced

equations

for curves of genus 2 with Jacobians of the specified RM.

References

1

Francois Morain.

Primality

Proving Using

Elliptic Curves

: An Update

. Springer: Berlin (2004)

2

Robert Nemiroff and Jerry Bonnell.

A million digits of sqrt(5)

at Project Gutenberg

http://www.gutenberg.org/dirs/etext96/5sqrt10.txt

http://www.gutenberg.org/dirs/etext96/5sqrt10.txt

3

Clifford Pickover.

Wonders of Numbers

, Oxford: Oxford University Press (2001) p. 106.

4

John Wilson “Curves of genus 2 with real

multiplication

by a square root of 5” p. i Dissertation, Oxford University, Oxford (1998)

http://eprints.maths.ox.ac.uk/32/01/wilson.pdf

http://eprints.maths.ox.ac.uk/32/01/wilson.pdf

Title

square root of 5

Canonical name

SquareRootOf5

Date of creation

2013-03-22 17:28:12

Last modified on

2013-03-22 17:28:12

Owner

MathNerd (17818)

Last modified by

MathNerd (17818)

Numerical id

6

Author

MathNerd (17818)

Entry type

Definition

Classification

msc 11A25

Generated on Fri Feb 9 15:14:56 2018 by

LaTeXML
square root of square root binomial

square root of square root binomial

Some people call the expressions of the form

a

+

b

⁢

c

the , especially when

c

is an square-free integer greater than 1 (and

a

and

b

rational numbers

).  On the high school (see e.g.

division

), or also

polynomials

containing several

square root

.  Taking the square root of a

square root binomial

is more difficult and usually results nested square roots.  However, there are some exceptions if the numbers are appropriate.  We have the

a

+

b

=

a

+

a

2

-

b

2

+

a

-

a

2

-

b

2

and

a

-

b

=

a

+

a

2

-

b

2

-

a

-

a

2

-

b

2

.

If

a

2

-

b

happens to be square of a rational number, then the into expressions without nested square roots.

For example, because

6

2

-

20

=

16

=

4

2

,  we obtain

6

+

2

⁢

5

=

6

+

20

=

6

+

4

2

+

6

-

4

2

=

1

+

5

,

and because

4

2

-

7

=

9

=

3

2

,  we get

4

-

7

=

4

+

3

2

-

4

-

3

2

=

7

-

1

2

=

14

-

2

2

.

References

1

K. Väisälä:

Algebran oppi- ja esimerkkikirja

I.   – Werner Söderström osakeyhtiö, Porvoo & Helsinki (1952).

Title

square root of square root binomial

Canonical name

SquareRootOfSquareRootBinomial

Date of creation

2013-03-22 15:21:21

Last modified on

2013-03-22 15:21:21

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Topic

Classification

msc 11A25

Related topic

TakingSquareRootAlgebraically

Related topic

SquareRootsOfRationals

Defines

square root binomial

Defines

square root polynomial

Generated on Fri Feb 9 15:14:59 2018 by

LaTeXML
sum of {μ⁢(n)}/n

sum of

μ

⁢

(

n

)

n

The following result holds:

∑

n

=

1

∞

μ

⁢

(

n

)

n

=

0

where

μ

⁢

(

n

)

is the Möbius

function

(

http://planetmath.org/MoebiusFunction

).

Proof:

Let

∑

n

=

1

∞

μ

⁢

(

n

)

n

=

α

. Assume

α

≠

0

.

For

Re

⁡

(

s

)

>

1

we have the

Euler product

expansion

1

ζ

⁢

(

s

)

=

∑

n

=

1

∞

μ

⁢

(

n

)

n

s

where

ζ

⁢

(

s

)

is the

Riemann zeta function

.

We recall the following

properties

of the Riemann zeta function (which can be found in the PlanetMath entry Riemann Zeta Function (

http://planetmath.org/RiemannZetaFunction

)).

•

ζ

⁢

(

s

)

is

analytic

except at the point

s

=

1

where it has a simple pole with

residue

1

.

•

ζ

⁢

(

s

)

has no zeroes in the region

Re

⁡

(

s

)

≥

1

.

•

The function

(

s

-

1

)

⁢

ζ

⁢

(

s

)

is analytic and nonzero for

Re

⁡

(

s

)

≥

1

.

•

Therefore, the function

1

ζ

⁢

(

s

)

is analytic for

Re

⁡

(

s

)

≥

1

.

Further, as a

corollary

of the proof of the

prime number theorem

, we also know that this sum,

∑

n

=

1

∞

μ

⁢

(

n

)

n

s

converges

to

1

ζ

⁢

(

s

)

for

Re

⁡

(

s

)

≥

1

; in particular, it converges

at

s

=

1

).

But then

ζ

⁢

(

1

)

=

1

∑

n

=

1

∞

μ

⁢

(

n

)

n

=

1

α

So

ζ

⁢

(

1

)

=

1

α

, but this is a

contradiction

since

ζ

has a simple pole at

s

=

1

. Therefore

α

=

0

.

Title

sum of

μ

⁢

(

n

)

n

Canonical

name

SumOffracmunn

Date of creation

2013-03-22 14:25:46

Last modified on

2013-03-22 14:25:46

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

13

Author

mathcam (2727)

Entry type

Result

Classification

msc 11A25

Related topic

MoebiusFunction

Generated on Fri Feb 9 15:15:03 2018 by

LaTeXML
summatory function of arithmetic function

summatory function of arithmetic function

Definition.

The

summatory function

F

of an

arithmetic function

f

is the Dirichlet convolution of

F

and the

constant function

1, i.e.

F

(

n

)

=

:

∑

d

∣

n

f

(

d

)

where

d

runs the

positive

divisors

of the

integer

n

.

It may be proved that the summatory function of a

multiplicative function

is multiplicative.

Theorem

.

The summatory function of the

Euler phi function

is the

identity function

:

∑

d

∣

n

φ

⁢

(

d

)

=

∑

d

∣

n

φ

⁢

(

n

d

)

=

n

for all

⁢

n

∈

ℤ

+

.

Proof.

The first equality follows from the fact that any positive divisor of

n

is got from

n

/

d

where

d

is a divisor of

n

.
Further, let

1

≤

m

≤

n

where

gcd

⁡

(

m

,

n

)

=

d

.  Then

gcd

⁡

(

m

/

d

,

n

/

d

)

=

1

and

1

≤

m

/

d

≤

n

/

d

.  This defines a

bijection

between the

prime classes

modulo

n

/

d

and such values of

m

in

{

1

,

2

,

…

,

n

-

1

}

for which

gcd

⁡

(

m

,

n

)

=

d

.  The number of the latters

φ

⁢

(

n

/

d

)

.
Furthermore, the only

m

with

1

≤

m

≤

n

and

gcd

⁡

(

m

,

n

)

=

n

is

m

:=

n

,  and

φ

⁢

(

n

/

n

)

=

φ

⁢

(

1

)

, by definition.

Summing

then over all possible values

d

yields the second equality.

References

1

Peter Hackman

:

Elementary number theory

.  HHH

productions

, Linköping (2009).

Title

summatory function of arithmetic function

Canonical name

SummatoryFunctionOfArithmeticFunction

Date of creation

2013-03-22 19:31:53

Last modified on

2013-03-22 19:31:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11A25

Synonym

summatory function

Related topic

EulerPhifunction

Related topic

PrimeClass

Generated on Fri Feb 9 15:15:05 2018 by

LaTeXML
superperfect number

superperfect number

A

k

-

superperfect number

n

is an

integer

such that

σ

k

⁢

(

n

)

=

2

⁢

n

, where

σ

k

⁢

(

x

)

is the

iterated sum of divisors function

. For example, 16 is 2-superperfect since its

divisors

add up to 31, and in turn the divisors of 31 add up to 32, which is twice 16.

At first Suryanarayana only considered 2-superperfect numbers. It is easy to prove that numbers of the form

2

p

-

1

are 2-superperfect only if

2

p

-

1

is a

Mersenne prime

. The existence of odd 2-superperfect numbers appears as unlikely as that of

regular

odd perfect numbers.

Later, Dieter Bode generalized the concept for any

k

and proved that when

k

>

2

there are no even

k

-superperfect numbers. Others have further generalized the concept to

(

k

,

m

)

-superperfect numbers satisifying the equality

σ

k

⁢

(

n

)

=

m

⁢

n

, and Weisstein programs a Mathematica command to default to

m

=

2

when the third argument is omitted. For example, 8, 21, and 512 are (2, 3)-superperfect, since the second iteration of the

sum of divisors function

gives thrice them, 24, 63, and 1536 respectively.

Not to be confused with

hyperperfect numbers

, which satisfy a different equality involving the sum of divisors function.

References

1

R. K. Guy,

Unsolved Problems in

Number Theory

New York: Springer-Verlag 2004: B9

2

D. Suryanarayana, “Super

perfect numbers

”

Elem. Math.

24

(1969): 16 - 17

3

E. Weisstein, “

http://mathworld.wolfram.com/SuperperfectNumber.html

Superperfect number”

Mathworld

Title

superperfect number

Canonical

name

SuperperfectNumber

Date of creation

2013-03-22 17:03:38

Last modified on

2013-03-22 17:03:38

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A25

Generated on Fri Feb 9 15:15:09 2018 by

LaTeXML
table of continued fractions of sqrt(n) for 1<n<102

table of continued fractions of

n

for

1

<

n

<

102

The

simple continued fractions

for the

square roots

of

positive integers

(which aren’t

perfect powers

) are non-terminating but they are periodic. In the following table, the square roots of the

integers

from 2 to 101 (excluding perfect powers) are listed in compact form: first the

integer part

followed by semicolon, then the periodic part stated once, its individual terms separated by commas. For example, the notation “14; 14, 28” for 198 means

198

=

14

+

1

14

+

1

28

+

1

14

+

1

28

+

…

,

where the dots mean a periodic repetition of 14 and 28 in the

denominators

.

n

Continued fraction

of

n

2

1; 2

3

1; 1, 2

5

2; 4

6

2; 2, 4

7

2; 1, 1, 1, 4

8

2; 1, 4

10

3; 6

11

3; 3, 6

12

3; 2, 6

13

3; 1, 1, 1, 1, 6

14

3; 1, 2, 1, 6

15

3; 1, 6

17

4; 8

18

4; 4, 8

19

4; 2, 1, 3, 1, 2, 8

20

4; 2, 8

21

4; 1, 1, 2, 1, 1, 8

22

4; 1, 2, 4, 2, 1, 8

23

4; 1, 3, 1, 8

24

4; 1, 8

26

5; 10

27

5; 5, 10

28

5; 3, 2, 3, 10

29

5; 2, 1, 1, 2, 10

30

5; 2, 10

31

5; 1, 1, 3, 5, 3, 1, 1, 10

32

5; 1, 1, 1, 10

33

5; 1, 2, 1, 10

34

5; 1, 4, 1, 10

35

5; 1, 10

37

6; 12

38

6; 6, 12

39

6; 4, 12

40

6; 3, 12

41

6; 2, 2, 12

42

6; 2, 12

43

6; 1, 1, 3, 1, 5, 1, 3, 1, 1, 12

44

6; 1, 1, 1, 2, 1, 1, 1, 12

45

6; 1, 2, 2, 2, 1, 12

46

6; 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12

47

6; 1, 5, 1, 12

48

6; 1, 12

50

7; 14

51

7; 7, 14

52

7; 4, 1, 2, 1, 4, 14

53

7; 3, 1, 1, 3, 14

54

7; 2, 1, 6, 1, 2, 14

55

7; 2, 2, 2, 14

56

7; 2, 14

57

7; 1, 1, 4, 1, 1, 14

58

7; 1, 1, 1, 1, 1, 1, 14

59

7; 1, 2, 7, 2, 1, 14

60

7; 1, 2, 1, 14

61

7; 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14

62

7; 1, 6, 1, 14

63

7; 1, 14

65

8; 16

66

8; 8, 16

67

8; 5, 2, 1, 1, 7, 1, 1, 2, 5, 16

68

8; 4, 16

69

8; 3, 3, 1, 4, 1, 3, 3, 16

70

8; 2, 1, 2, 1, 2, 16

71

8; 2, 2, 1, 7, 1, 2, 2, 16

72

8; 2, 16

73

8; 1, 1, 5, 5, 1, 1, 16

74

8; 1, 1, 1, 1, 16

75

8; 1, 1, 1, 16

76

8; 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16

77

8; 1, 3, 2, 3, 1, 16

78

8; 1, 4, 1, 16

79

8; 1, 7, 1, 16

80

8; 1, 16

82

9; 18

83

9; 9, 18

84

9; 6, 18

85

9; 4, 1, 1, 4, 18

86

9; 3, 1, 1, 1, 8, 1, 1, 1, 3, 18

87

9; 3, 18

88

9; 2, 1, 1, 1, 2, 18

89

9; 2, 3, 3, 2, 18

90

9; 2, 18

91

9; 1, 1, 5, 1, 5, 1, 1, 18

92

9; 1, 1, 2, 4, 2, 1, 1, 18

93

9; 1, 1, 1, 4, 6, 4, 1, 1, 1, 18

94

9; 1, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, 3, 2, 1, 18

95

9; 1, 2, 1, 18

96

9; 1, 3, 1, 18

97

9; 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18

98

9; 1, 8, 1, 18

99

9; 1, 18

101

10; 20

As the table shows, the periodic part ends with

2

⁢

⌊

n

⌋

.

Title

table of continued fractions of

n

for

1

<

n

<

102

Canonical name

TableOfContinuedFractionsOfsqrtnFor1N102

Date of creation

2013-03-22 17:30:25

Last modified on

2013-03-22 17:30:25

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11A25

Generated on Fri Feb 9 15:15:11 2018 by

LaTeXML
table of differences between ERROR(\delimiter)⁢"⁢4264306⁢sqrt(n!)⁢ERROR(\delimiter)⁢"⁢5265307^2 and n! for 0<n<26

table of differences between

\delimiter

⁢

"

⁢

4264306

⁢

n

!

⁢

\delimiter

⁢

"

⁢

5265307

2

and

n

!

for

0

<

n

<

26

There are only three known solutions to Brocard’s problem, and the near misses all seem to occur early on. Notice how, for example, 3! is just 3 shy of a square (compared to 1 shy of a square which is what Brocard’s problem asks for). Still, the

differences

between a

factorial

and the next higher

perfect square

don’t make for a consistently

ascending order

sequence

. For a few values of

n

, (such as 4, 7, 10, 24, 26, 42, 117, 135) this difference is smaller than the previous difference. In general, however, the difference between a factorial and the next perfect square widens as

n

gets larger.

The following table gives the

square root

of

n

!

to six decimal places, and then the difference between the factorial and the next higher square (obtained by taking the

ceiling

of the square root of

n

!

and squaring that

integer

).

n

n

!

⌈

n

!

⌉

2

-

n

!

1

1.000000

0

2

1.414214

2

3

2.449489

3

4

4.898979

1

5

10.954451

1

6

26.832816

9

7

70.992957

1

8

200.798406

81

9

602.395219

729

10

1904.940944

225

11

6317.974359

324

12

21886.105181

39169

13

78911.474451

82944

14

295259.701280

176400

15

1143535.905864

215296

16

4574143.623456

3444736

17

18859677.306253

26167684

18

80014834.285449

114349225

19

348776576.634429

255004929

20

1559776268.628498

1158920361

21

7147792818.185865

11638526761

22

33526120082.371712

42128246889

23

160785623545.405884

191052974116

24

787685471322.938354

97216010329

25

3938427356614.691406

2430400258225

Title

table of differences between

\delimiter

⁢

"

⁢

4264306

⁢

n

!

⁢

\delimiter

⁢

"

⁢

5265307

2

and

n

!

for

0

<

n

<

26

Canonical name

TableOfDifferencesBetweenlceilsqrtnrceil2AndNFor0N26

Date of creation

2013-03-22 18:10:20

Last modified on

2013-03-22 18:10:20

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11A25

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

LaTeXML
table of values of the Liouville function and its summatory function

table of values of the Liouville function and its summatory function

The following table lists the values of the

Liouville function

λ

⁢

(

n

)

and the summatory Liouville function

L

⁢

(

n

)

for

0

<

n

<

101

. The Liouville function is defined as

λ

⁢

(

n

)

=

(

-

1

)

Ω

⁢

(

n

)

(where

Ω

⁢

(

n

)

is the number of nondistinct

prime factors

function

). The

matching

summatory function

for the Liouville function is

L

⁢

(

n

)

=

∑

i

=

1

n

λ

⁢

(

i

)

.

n

λ

⁢

(

n

)

L

⁢

(

n

)

n

λ

⁢

(

n

)

L

⁢

(

n

)

n

λ

⁢

(

n

)

L

⁢

(

n

)

n

λ

⁢

(

n

)

L

⁢

(

n

)

1

1

1

26

1

0

51

1

-

5

76

-

1

-

8

2

-

1

0

27

-

1

-

1

52

-

1

-

6

77

1

-

7

3

-

1

-

1

28

-

1

-

2

53

-

1

-

7

78

-

1

-

8

4

1

0

29

-

1

-

3

54

1

-

6

79

-

1

-

9

5

-

1

-

1

30

-

1

-

4

55

1

-

5

80

-

1

-1

0

6

1

0

31

-

1

-

5

56

1

-

4

81

1

-

9

7

-

1

-

1

32

-

1

-

6

57

1

-

3

82

1

-

8

8

-

1

-

2

33

1

-

5

58

1

-

2

83

-

1

-

9

9

1

-

1

34

1

-

4

59

-

1

-

3

84

1

-

8

10

1

0

35

1

-

3

60

1

-

2

85

1

-

7

11

-

1

-

1

36

1

-

2

61

-

1

-

3

86

1

-

6

12

-

1

-

2

37

-

1

-

3

62

1

-

2

87

1

-

5

13

-

1

-

3

38

1

-

2

63

-

1

-

3

88

1

-

4

14

1

-

2

39

1

-

1

64

1

-

2

89

-

1

-

5

15

1

-

1

40

1

0

65

1

-

1

90

1

-

4

16

1

0

41

-

1

-

1

66

-

1

-

2

91

1

-

3

17

-

1

-

1

42

-

1

-

2

67

-

1

-

3

92

-

1

-

4

18

-

1

-

2

43

-

1

-

3

68

-

1

-

4

93

1

-

3

19

-

1

-

3

44

-

1

-

4

69

1

-

3

94

1

-

2

20

-

1

-

4

45

-

1

-

5

70

-

1

-

4

95

1

-

1

21

1

-

3

46

1

-

4

71

-

1

-

5

96

1

0

22

1

-

2

47

-

1

-

5

72

-

1

-

6

97

-

1

-

1

23

-

1

-

3

48

-

1

-

6

73

-

1

-

7

98

-

1

-

2

24

1

-

2

49

1

-

5

74

1

-

6

99

-

1

-

3

25

1

-

1

50

-

1

-

6

75

-

1

-

7

100

1

-

2

Title

table of values of the Liouville function and its summatory function

Canonical name

TableOfValuesOfTheLiouvilleFunctionAndItsSummatoryFunction

Date of creation

2013-03-22 18:06:45

Last modified on

2013-03-22 18:06:45

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11A25

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

LaTeXML
table of values of the Möbius function and the Mertens function

table of values of the Möbius function and the Mertens function

The following table lists the values of the Möbius function

μ

⁢

(

n

)

and the

Mertens function

M

⁢

(

n

)

for

0

<

n

<

101

. The Möbius function is defined as

μ

⁢

(

n

)

=

(

-

1

)

ω

⁢

(

n

)

(where

ω

⁢

(

n

)

is the

number of distinct prime factors function

) for

squarefree

numbers, and

μ

⁢

(

n

)

=

0

for any

integer

with a repeated

prime factor

. The Mertens function is the

matching

summatory function

for the Möbius function,

M

⁢

(

n

)

=

∑

i

=

1

n

μ

⁢

(

i

)

.

n

μ

⁢

(

n

)

M

⁢

(

n

)

n

μ

⁢

(

n

)

M

⁢

(

n

)

n

μ

⁢

(

n

)

M

⁢

(

n

)

n

μ

⁢

(

n

)

M

⁢

(

n

)

1

1

1

26

1

-

1

51

1

-

2

76

0

-

3

2

-

1

0

27

0

-

1

52

0

-

2

77

1

-

2

3

-

1

-

1

28

0

-

1

53

-

1

-

3

78

-

1

-

3

4

0

-

1

29

-

1

-

2

54

0

-

3

79

-

1

-

4

5

-

1

-

2

30

-

1

-

3

55

1

-

2

80

0

-

4

6

1

-

1

31

-

1

-

4

56

0

-

2

81

0

-

4

7

-

1

-

2

32

0

-

4

57

1

-

1

82

1

-

3

8

0

-

2

33

1

-

3

58

1

0

83

-

1

-

4

9

0

-

2

34

1

-

2

59

-

1

-

1

84

0

-

4

10

1

-

1

35

1

-

1

60

0

-

1

85

1

-

3

11

-

1

-

2

36

0

-

1

61

-

1

-

2

86

1

-

2

12

0

-

2

37

-

1

-

2

62

1

-

1

87

1

-

1

13

-

1

-

3

38

1

-

1

63

0

-

1

88

0

-

1

14

1

-

2

39

1

0

64

0

-

1

89

-

1

-

2

15

1

-

1

40

0

0

65

1

0

90

0

-

2

16

0

-

1

41

-

1

-

1

66

-

1

-

1

91

1

-

1

17

-

1

-

2

42

-

1

-

2

67

-

1

-

2

92

0

-

1

18

0

-

2

43

-

1

-

3

68

0

-

2

93

1

0

19

-

1

-

3

44

0

-

3

69

1

-

1

94

1

1

20

0

-

3

45

0

-

3

70

-

1

-

2

95

1

2

21

1

-

2

46

1

-

2

71

-

1

-

3

96

0

2

22

1

-

1

47

-

1

-

3

72

0

-

3

97

-

1

1

23

-

1

-

2

48

0

-

3

73

-

1

-

4

98

0

1

24

0

-

2

49

0

-

3

74

1

-

3

99

0

1

25

0

-

2

50

0

-

3

75

0

-

3

100

0

1

Title

table of values of the Möbius function and the Mertens function

Canonical name

TableOfValuesOfTheMobiusFunctionAndTheMertensFunction

Date of creation

2013-03-22 18:06:27

Last modified on

2013-03-22 18:06:27

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11A25

Generated on Fri Feb 9 15:15:20 2018 by

LaTeXML
the sum of the values of a character of a finite group is 0

the sum of the values of a character of a finite group is

0

The following is an

argument

that

occurs in

many proofs involving characters of groups. Here we use additive notation for the group

G

, however this group is not assumed to be

abelian

.

Lemma 1

.

Let

G

be a

finite group

, and let

K

be a field. Let

χ

:

G

→

K

×

be a

character

, where

K

×

denotes the

multiplicative group

of

K

. Then:

∑

g

∈

G

χ

⁢

(

g

)

=

{

∣

G

∣

,

if

⁢

χ

⁢

is trivial,

0

K

,

otherwise

where

0

K

is the

zero element

in

K

, and

∣

G

∣

is the order of the group

G

.

Proof.

First assume that

χ

is trivial, i.e. for all

g

∈

G

we have

χ

⁢

(

g

)

=

1

∈

K

. Then the result is clear.

Thus, let us assume that there exists

g

1

in

G

such that

χ

⁢

(

g

1

)

=

h

≠

1

∈

K

. Notice that for any element

g

1

∈

G

the map:

G

→

G

,

g

↦

g

1

+

g

is clearly a bijection. Define

𝒮

=

∑

g

∈

G

χ

⁢

(

g

)

∈

K

. Then:

h

⋅

𝒮

=

χ

⁢

(

g

1

)

⋅

𝒮

=

χ

⁢

(

g

1

)

⋅

∑

g

∈

G

χ

⁢

(

g

)

=

∑

g

∈

G

χ

⁢

(

g

1

)

⋅

χ

⁢

(

g

)

=

∑

g

∈

G

χ

⁢

(

g

1

+

g

)

,

(

1

)

=

∑

j

∈

G

χ

⁢

(

j

)

,

(

2

)

=

𝒮

By the remark above, sums

(

1

)

and

(

2

)

are equal, since both run over all possible values of

χ

over elements of

G

. Thus, we have proved that:

h

⋅

𝒮

=

𝒮

and

h

≠

1

∈

K

. Since

K

is a field, it follows that

𝒮

=

0

∈

K

, as desired.

∎

Title

the sum of the values of a character of a finite group is

0

Canonical name

TheSumOfTheValuesOfACharacterOfAFiniteGroupIs0

Date of creation

2013-03-22 14:10:30

Last modified on

2013-03-22 14:10:30

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11A25

Generated on Fri Feb 9 15:15:22 2018 by

LaTeXML