Dedekind Eta Function (original) (raw)

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DedekindEta

The Dedekind eta function is defined over the upper half-plane H={tau:I[tau]>0} by

(OEIS A010815), where q^_=e^(2piitau) is the square of the nome , tau is the half-period ratio, and (q)_infty 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 q^_ explicitly in terms of the half-period ratio tau gives the product

	(7)

DedekindEtaReIm

DedekindEtaContours

It is illustrated above in the complex plane.

eta(tau) 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 zeta(z;g_2,g_3) is the Weierstrass zeta function and g_2 and g_3 are the invariants corresponding to the half-periods (1,tau) . The derivative of eta(tau) satisfies

	(10)

where G_2(tau) is an Eisenstein series, and

	(11)

A special value is given by

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

where is the plastic constant, (P(x))_n denotes a polynomial root, and tau_0=(1+isqrt(23))/2 .

Letting zeta_(24)=e^(2pii/24)=e^(pii/12) be a root of unity, eta(tau) satisfies

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 theta_2 by

	(20)

(Weber 1902, Vol. 3, p. 112) and

	(21)

(Apostol 1997, p. 91).

Macdonald (1972) has related most expansions of the form (q,q)_infty^c to affine root systems. Exceptions not included in Macdonald's treatment include c=2 , found by Hecke and Rogers, c=4 , found by Ramanujan, and c=26 , 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 [a b; c d] in Gamma , where Gamma is the modular group Gamma, c>0 , and tau in H (where is the upper half-plane), then

	(24)

where

	(25)

and

| $s(h,k)=sum_(r=1)^(k-1)r/k((hr)/k-\|_(hr)/k_|-1/2)$ | (26) | | ------------------------------------------------------------------------------------------------------------------------------------------------- | ---- |

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

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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 _1psi_1 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 (q)_infty^(n^2+n) and Basic Hypergeometric Series in U(n) ." Discr. Math. 204, 281-317, 1999a.Leininger, V. E. and Milne, S. C. "Some New Infinite Families of eta -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 eta -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 eta(-1/tau)=eta(tau)sqrt(tau/i) ." 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.