Euler function

Properties

The coefficient p ( k ) in the formal power series expansion for 1 / ϕ ( q ) gives the number of all partitions of k. That is,

1 ϕ ( q ) = ∑ k = 0 ∞ p ( k ) q k

where p ( k ) is the partition function of k.

The Euler identity, also known as the Pentagonal number theorem is

ϕ ( q ) = ∑ n = − ∞ ∞ ( − 1 ) n q ( 3 n 2 − n ) / 2 .

Note that ( 3 n 2 − n ) / 2 is a pentagonal number.

The Euler function is related to the Dedekind eta function through a Ramanujan identity as

ϕ ( q ) = q − 1 24 η ( τ )

where q = e 2 π i τ is the square of the nome.

Note that both functions have the symmetry of the modular group.

The Euler function may be expressed as a Q-Pochhammer symbol:

ϕ ( q ) = ( q ; q ) ∞

The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q=0, yielding:

ln ⁡ ( ϕ ( q ) ) = − ∑ n = 1 ∞ 1 n q n 1 − q n

which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as:

ln ⁡ ( ϕ ( q ) ) = ∑ n = 1 ∞ b n q n

where

b n = − ∑ d | n 1 d = -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] (see OEIS A000203)

On account of the following identity,

∑ d | n d = ∑ d | n n d

this may also be written as

ln ⁡ ( ϕ ( q ) ) = − ∑ n = 1 ∞ q n n ∑ d | n d

Special values

The next identities come from Ramanujan's lost notebook, Part V, p. 326.

ϕ ( e − π ) = e π / 24 Γ ( 1 4 ) 2 7 / 8 π 3 / 4 ϕ ( e − 2 π ) = e π / 12 Γ ( 1 4 ) 2 π 3 / 4 ϕ ( e − 4 π ) = e π / 6 Γ ( 1 4 ) 2 11 / 8 π 3 / 4 ϕ ( e − 8 π ) = e π / 3 Γ ( 1 4 ) 2 29 / 16 π 3 / 4 ( 2 − 1 ) 1 / 4