Dedekind Eta Function

Min Max

The Dedekind eta function is defined over the upper half-plane  by

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)

(OEIS A010815), where  is the square of the nome ,  is the half-period ratio, and  is a q-series (Weber 1902, pp. 85 and 112; Atkin and Morain 1993; Berndt 1994, p. 139).

The Dedekind eta function is implemented in the Wolfram Language as DedekindEta[tau].

Rewriting the definition in terms of  explicitly in terms of the half-period ratio  gives the product

(7)

Min   Max       Re       Im

It is illustrated above in the complex plane.

 is a modular form first introduced by Dedekind in 1877, and is related to the modular discriminant of the Weierstrass elliptic function by

(8)

(Apostol 1997, p. 47).

A compact closed form for the derivative is given by

(9)

where  is the Weierstrass zeta function and  and  are the invariants corresponding to the half-periods . The derivative of  satisfies

(10)

where  is an Eisenstein series, and

(11)

A special value is given by

			(12)
			(13)

(OEIS A091343), where  is the gamma function. Another special case is

			(14)
			(15)
			(16)

where  is the plastic constant,  denotes a polynomial root, and .

Letting  be a root of unity,  satisfies

			(17)
			(18)
			(19)

where  is an integer (Weber 1902, p. 113; Atkin and Morain 1993; Apostol 1997, p. 47). The Dedekind eta function is related to the Jacobi theta function  by

(20)

(Weber 1902, Vol. 3, p. 112) and

(21)

(Apostol 1997, p. 91).

Macdonald (1972) has related most expansions of the form  to affine root systems. Exceptions not included in Macdonald's treatment include , found by Hecke and Rogers, , found by Ramanujan, and , found by Atkin (Leininger and Milne 1999). Using the Dedekind eta function, the Jacobi triple product identity

(22)

can be written

(23)

(Jacobi 1829, Hardy and Wright 1979, Hirschhorn 1999, Leininger and Milne 1999).

Dedekind's functional equation states that if , where  is the modular group Gamma, , and  (where  is the upper half-plane), then

(24)

where

(25)

and

(26)

is a Dedekind sum (Apostol 1997, pp. 52-57), with  the floor function.

REFERENCES:

Apostol, T. M. "The Dedekind Eta Function." Ch. 3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 47-73, 1997.

Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primality Proving." Math. Comput. 61, 29-68, 1993.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.

Bhargava, S. and Somashekara, D. "Some Eta-Function Identities Deducible from Ramanujan's  Summation." J. Math. Anal. Appl. 176, 554-560, 1993.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Hirschhorn, M. D. "Another Short Proof of Ramanujan's Mod 5 Partition Congruences, and More." Amer. Math. Monthly 106, 580-583, 1999.

Jacobi, C. G. J. Fundamenta Nova Theoriae Functionum Ellipticarum. Königsberg, Germany: Regiomonti, Sumtibus fratrum Borntraeger, p. 90, 1829.

Leininger, V. E. and Milne, S. C. "Expansions for  and Basic Hypergeometric Series in ." Discr. Math. 204, 281-317, 1999a.

Leininger, V. E. and Milne, S. C. "Some New Infinite Families of -Function Identities." Methods Appl. Anal. 6, 225-248, 1999b.

Köhler, G. "Some Eta-Identities Arising from Theta Series." Math. Scand. 66, 147-154, 1990.

Macdonald, I. G. "Affine Root Systems and Dedekind's -Function." Invent. Math. 15, 91-143, 1972.

Ramanujan, S. "On Certain Arithmetical Functions." Trans. Cambridge Philos. Soc. 22, 159-184, 1916.

Siegel, C. L. "A Simple Proof of ." Mathematika 1, 4, 1954.

Sloane, N. J. A. Sequences A010815, A091343, and A116397 in "The On-Line Encyclopedia of Integer Sequences."

Weber, H. Lehrbuch der Algebra, Vols. I-III. 1902. Reprinted as Lehrbuch der Algebra, Vols. I-III, 3rd rev ed. New York: Chelsea, 1979.