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| Dedekind eta function | |
|---|---|
| Name | Dedekind eta function |
| Domain | Upper half-plane |
| Introduced | 1877 |
| Introduced by | Richard Dedekind |
| Related | Modular discriminant, Jacobi theta functions, Ramanujan tau function |
Dedekind eta function The Dedekind eta function is a holomorphic function on the complex upper half-plane introduced by Richard Dedekind in 1877 and central to the theory developed by Bernhard Riemann, Felix Klein, Henri Poincaré, and Srinivasa Ramanujan. It links the work of Leopold Kronecker, Karl Weierstrass, David Hilbert, Émile Picard, and Heinrich Weber to modern developments by André Weil, John Tate, Pierre Deligne, and Jean-Pierre Serre. The eta function plays a key role in the theories advanced at institutions such as University of Göttingen, École Normale Supérieure, and Princeton University through connections with the Modular group, Elliptic curves, and Monstrous moonshine.
The Dedekind eta function is defined for τ in the complex upper half-plane by η(τ)=e^{π i τ/12} ∏_{n=1}^∞ (1−e^{2π i n τ}), reflecting constructions in Fourier analysis used by Peter Gustav Lejeune Dirichlet and techniques employed by Karl Pearson and Godfrey Harold Hardy. As a holomorphic function it transforms under the Modular group generated by transformations considered by Émile Picard and Henri Poincaré and satisfies functional relations analogous to those studied by Adrien-Marie Legendre and Niels Henrik Abel. The eta function is nonvanishing on the upper half-plane except for controlled behavior at cusps studied by John von Neumann and Atle Selberg.
Under τ↦τ+1 and τ↦−1/τ the eta function obeys a multiplier system related to the Modular group and the two generators associated to work by Felix Klein and Henri Poincaré. Its transformation law involves twelfth roots of unity connected to computations by Richard Dedekind and congruences studied by Ernst Kummer and Srinivasa Ramanujan. The automorphy factor for η arises in contexts examined by Emil Artin, André Weil, Serge Lang, and I. M. Gelfand when relating η to modular forms on congruence subgroups invoked in theorems by Atkin and Lehner.
The q-expansion η(τ)=q^{1/24} ∏_{n≥1} (1−q^n) with q=e^{2π i τ} is foundational in expansions used by Ramanujan, G. H. Hardy, John Littlewood, and George Pólya. The Eulerian infinite product mirrors identities studied by Leonhard Euler and Jacques Hadamard and is connected to partition generating functions investigated by Harry Andrews, Freeman Dyson, and Ken Ono. Variants of the product appear in identities of Carl Gustav Jacobi for theta functions and in transformation formulae exploited by Erich Hecke in his operator theory.
The eta function has no zeros in the upper half-plane, with its only zeros at cusps such as infinity; these properties are treated using the valence formula formalized by Heinrich Matthias, concepts related to the classification by Bernhard Riemann and applications by Atle Selberg. Orders of vanishing at cusps are central to results by Martin Eichler and Don Zagier and to congruence phenomena studied by Nicholas Katz and Jean-Pierre Serre. Analyses of growth, zero-free regions, and distribution draw on spectral ideas from Harish-Chandra, Ilya Piatetski-Shapiro, and Robert Langlands.
The twelfth power η(τ)^{24} is proportional to the modular discriminant Δ(τ), an object underlying the work of Bernhard Riemann on zeta functions and of Alexander Grothendieck on schemes; Δ features in the formulation of the Taniyama–Shimura–Weil conjecture proved by Andrew Wiles and Richard Taylor. Relations between η and classical Eisenstein series studied by Gotthold Eisenstein and Srinivasa Ramanujan link to Hecke operators developed by Erich Hecke and to Galois representations investigated by Jean-Pierre Serre and Richard Taylor. The connection to the j-invariant used by Kurt Heegner and Alan Baker situates η at the heart of the arithmetic of Elliptic curves and complex multiplication studied by Henri Cohen and Goro Shimura.
Eta-based q-series underpin partition congruences discovered by Srinivasa Ramanujan and later explored by Ken Ono, George Andrews, and Bruce Berndt. Modular transformations of η inform proofs in the theory of class numbers pursued by Dirichlet and Heegner and appear in research on L-functions by Atle Selberg and Goro Shimura. Combinatorial identities derived from η-products have been applied in work at Institute for Advanced Study and in enumerative problems relevant to the Monstrous moonshine conjectures proved by Richard Borcherds.
Generalizations include eta-products and eta-quotients analyzed by Hecke, Atkin, and Newman; vector-valued analogues studied by Borcherds and James Lepowsky; and connections to Jacobi theta functions developed by Carl Gustav Jacobi and furthered by Dmitry Zagier. Higher-level analogues and multiplier systems relate to the representation theory treated by Harish-Chandra and to conformal field theory frameworks at CERN and in work by Edward Witten. Recent developments tie eta-generalizations to moonshine phenomena explored by John Conway, Simon Norton, and Ching Hung Lam.