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