Euler function

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In mathematics, the Euler function is given by

ϕ ( q ) = ∏ k = 1 ∞ ( 1 − q k ) , | q | < 1. {\displaystyle \phi (q)=\prod _{k=1}^{\infty }(1-q^{k}),\quad |q|<1.}

Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics and complex analysis.

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

1 ϕ ( q ) = ∑ k = 0 ∞ p ( k ) q k {\displaystyle {\frac {1}{\phi (q)}}=\sum _{k=0}^{\infty }p(k)q^{k}}

where p {\displaystyle p} is the partition function.

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

ϕ ( q ) = ∑ n = − ∞ ∞ ( − 1 ) n q ( 3 n 2 − n ) / 2 . {\displaystyle \phi (q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(3n^{2}-n)/2}.}

( 3 n 2 − n ) / 2 {\displaystyle (3n^{2}-n)/2} is a pentagonal number.

The Euler function is related to the Dedekind eta function as

ϕ ( e 2 π i τ ) = e − π i τ / 12 η ( τ ) . {\displaystyle \phi (e^{2\pi i\tau })=e^{-\pi i\tau /12}\eta (\tau ).}

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

ϕ ( q ) = ( q ; q ) ∞ . {\displaystyle \phi (q)=(q;q)_{\infty }.}

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 , {\displaystyle \ln(\phi (q))=-\sum _{n=1}^{\infty }{\frac {1}{n}}\,{\frac {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 {\displaystyle \ln(\phi (q))=\sum _{n=1}^{\infty }b_{n}q^{n}}

where b n = − ∑ d | n 1 d = {\displaystyle b_{n}=-\sum _{d|n}{\frac {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 identity σ ( n ) = ∑ d | n d = ∑ d | n n d {\displaystyle \sigma (n)=\sum _{d|n}d=\sum _{d|n}{\frac {n}{d}}} , where σ ( n ) {\displaystyle \sigma (n)} is the sum-of-divisors function, this may also be written as

ln ⁡ ( ϕ ( q ) ) = − ∑ n = 1 ∞ σ ( n ) n   q n {\displaystyle \ln(\phi (q))=-\sum _{n=1}^{\infty }{\frac {\sigma (n)}{n}}\ q^{n}} .

Also if a , b ∈ R + {\displaystyle a,b\in \mathbb {R} ^{+}} and a b = π 2 {\displaystyle ab=\pi ^{2}} , then^[1]

a 1 / 4 e − a / 12 ϕ ( e − 2 a ) = b 1 / 4 e − b / 12 ϕ ( e − 2 b ) . {\displaystyle a^{1/4}e^{-a/12}\phi (e^{-2a})=b^{1/4}e^{-b/12}\phi (e^{-2b}).}

The next identities come from Ramanujan's Notebooks:^[2]

ϕ ( e − π ) = e π / 24 Γ ( 1 4 ) 2 7 / 8 π 3 / 4 {\displaystyle \phi (e^{-\pi })={\frac {e^{\pi /24}\Gamma \left({\frac {1}{4}}\right)}{2^{7/8}\pi ^{3/4}}}}

ϕ ( e − 2 π ) = e π / 12 Γ ( 1 4 ) 2 π 3 / 4 {\displaystyle \phi (e^{-2\pi })={\frac {e^{\pi /12}\Gamma \left({\frac {1}{4}}\right)}{2\pi ^{3/4}}}}

ϕ ( e − 4 π ) = e π / 6 Γ ( 1 4 ) 2 11 / 8 π 3 / 4 {\displaystyle \phi (e^{-4\pi })={\frac {e^{\pi /6}\Gamma \left({\frac {1}{4}}\right)}{2^{{11}/8}\pi ^{3/4}}}}

ϕ ( e − 8 π ) = e π / 3 Γ ( 1 4 ) 2 29 / 16 π 3 / 4 ( 2 − 1 ) 1 / 4 {\displaystyle \phi (e^{-8\pi })={\frac {e^{\pi /3}\Gamma \left({\frac {1}{4}}\right)}{2^{29/16}\pi ^{3/4}}}({\sqrt {2}}-1)^{1/4}}

Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives^[3]

∫ 0 1 ϕ ( q ) d q = 8 3 23 π sinh ⁡ ( 23 π 6 ) 2 cosh ⁡ ( 23 π 3 ) − 1 . {\displaystyle \int _{0}^{1}\phi (q)\,\mathrm {d} q={\frac {8{\sqrt {\frac {3}{23}}}\pi \sinh \left({\frac {{\sqrt {23}}\pi }{6}}\right)}{2\cosh \left({\frac {{\sqrt {23}}\pi }{3}}\right)-1}}.}

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001