Ramanujan tau function

Ramanujan tau function

The

Ramanujan tau function

is the

arithmetic function

τ

:

ℕ

→

ℤ

such that, for all

q

∈

ℂ

with

|

q

|

<

1

,

q

⁢

∏

k

=

1

∞

(

1

-

q

k

)

24

=

∑

n

=

1

∞

τ

⁢

(

n

)

⁢

q

n

.

Thus, the Ramanujan tau function is the

generating function

for the Weierstrass

Δ

function

(

http://

planetmath

.org/ModularForms

).

Determining values of the Ramanujan tau function directly can be somewhat involved. For example, the values of

τ

⁢

(

1

)

,

τ

⁢

(

2

)

, and

τ

⁢

(

3

)

will be determined:

To determine

τ

⁢

(

1

)

,

τ

⁢

(

2

)

, and

τ

⁢

(

3

)

, we need to find the coefficient of

q

,

q

2

, and

q

3

, respectively, of the

expression

q

⁢

∏

k

=

1

∞

(

1

-

q

k

)

24

.

Note that we only need to consider

k

=

1

and

k

=

2

, since higher values of

k

yield powers (

http://planetmath.org/Power

) of

q

that are too large. Thus:

q

⁢

(

1

-

q

)

24

⁢

(

1

-

q

2

)

24

=

q

⁢

(

1

-

24

⁢

q

+

276

⁢

q

2

-

…

)

⁢

(

1

-

24

⁢

q

2

+

…

)

=

q

⁢

(

1

-

24

⁢

q

+

276

⁢

q

2

-

…

-

24

⁢

q

2

+

576

⁢

q

3

-

…

)

=

q

⁢

(

1

-

24

⁢

q

+

252

⁢

q

2

-

…

)

=

q

-

24

⁢

q

2

+

252

⁢

q

3

-

…

Hence,

τ

⁢

(

1

)

=

1

,

τ

⁢

(

2

)

=

-

24

, and

τ

⁢

(

3

)

=

252

.

The

sequence

{

τ

⁢

(

n

)

}

appears in the OEIS as sequence

http://www.research.att.com/ njas/sequences/A000594

A000594.

Although the values of

|

τ

⁢

(

n

)

|

seem to increase rapidly as

n

increases, the

conjecture

that

τ

⁢

(

n

)

≠

0

for all

n

∈

ℕ

has not yet been proven. This conjecture is known as

Lehmer’s conjecture

.

The Ramanujan tau function has the following

properties

:

•

It is a

multiplicative function

: For

a

,

b

∈

ℕ

with

gcd

⁡

(

a

,

b

)

=

1

, we have

τ

⁢

(

a

⁢

b

)

=

τ

⁢

(

a

)

⁢

τ

⁢

(

b

)

.

•

For any prime

p

and any

n

∈

ℕ

,

τ

⁢

(

p

n

+

1

)

=

τ

⁢

(

p

)

⁢

τ

⁢

(

p

n

)

-

p

11

⁢

τ

⁢

(

p

n

-

1

)

.

•

For any prime

p

,

|

τ

⁢

(

p

)

|

≤

2

⁢

p

11

2

.

Ramanujan

asserted that

τ

several

congruences

, all of which have been proven. Some simpler examples of such congruences include:

•

For any

n

∈

ℕ

,

τ

⁢

(

5

⁢

n

)

≡

0

(

mod

5

)

.

•

For any

n

∈

ℕ

and for any

nonnegative integer

r

<

7

which is a

quadratic residue

modulo

7

,

τ

⁢

(

7

⁢

n

-

r

)

≡

0

(

mod

7

)

.

•

For any

n

∈

ℕ

and for any nonnegative integer

r

<

23

which is a quadratic residue modulo

23

,

τ

⁢

(

23

⁢

n

-

r

)

≡

0

(

mod

23

)

.

References

1

Berndt, Bruce C.

Number Theory

in the Spirit of Ramanujan

. Providence, RI:

American Mathematical Society

, 2006.

Title

Ramanujan tau function

Canonical

name

RamanujanTauFunction

Date of creation

2013-03-22 17:51:24

Last modified on

2013-03-22 17:51:24

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

12

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11F11

Classification

msc 11A25

Synonym

Ramanujan’s tau function

Related topic

ModularForms

Related topic

ModularDiscriminant

Related topic

Ramanujan

Related topic

ApplicationsOfSecondOrderRecurrenceRelationFormula

Defines

Lehmer’s conjecture

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