In mathematics, the Euler function is given by

Domain coloring plot of ϕ on the complex plane

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.

Properties

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The coefficient   in the formal power series expansion for   gives the number of partitions of k. That is,

 

where   is the partition function.

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

 

  is a pentagonal number.

The Euler function is related to the Dedekind eta function as

 

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

 

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

 

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

 

where   -[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   , where   is the sum-of-divisors function, this may also be written as

 .

Also if   and  , then[1]

 

Special values

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

 
 
 
 

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

 

References

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  1. ^ Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction"
  2. ^ Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V. Springer. ISBN 978-1-4612-7221-2. p. 326
  3. ^ Sloane, N. J. A. (ed.). "Sequence A258232". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.