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Difference between revisions of "Dedekind eta-function"

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The function defined by
 
The function defined by
  
 
$$\eta(z)=e^{\pi iz/12}\prod_{n=1}^\infty(1-e^{2\pi inz})$$
 
$$\eta(z)=e^{\pi iz/12}\prod_{n=1}^\infty(1-e^{2\pi inz})$$
  
for $z\in\mathbf C$, $\operatorname{Im}z>0$. As the [[Infinite product|infinite product]] converges absolutely, uniformly for $z$ in compact sets (cf. [[Uniform convergence|Uniform convergence]]), the function $\eta$ is holomorphic (cf. [[Analytic function|Analytic function]]). Moreover, it satisfies $\eta(z+1)=e^{\pi i/12}\eta(z)$ and $\eta(-1/z)=\sqrt{-iz}\eta(z)$. So, $\eta^24$ is a [[Modular form|modular form]] of weight $12$ (cf. also [[Modular group|Modular group]]).
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for $z\in\mathbf C$, $\operatorname{Im}z>0$. As the [[infinite product]] converges absolutely, uniformly for $z$ in compact sets (cf. [[Uniform convergence]]), the function $\eta$ is holomorphic (cf. [[Analytic function]]). Moreover, it satisfies $\eta(z+1)=e^{\pi i/12}\eta(z)$ and $\eta(-1/z)=\sqrt{-iz}\eta(z)$. So, $\eta^{24}$ is a [[modular form]] of weight $12$ (cf. also [[Modular group]]).
  
R. Dedekind [[#References|[a1]]] comments on computations of B. Riemann in connection with theta-functions (cf. [[Theta-function|Theta-function]]). He shows that it is basic to understand the transformation behaviour of the logarithm of the function now carrying his name. This study leads him to quantities now called Dedekind sums (cf. [[Dedekind sum|Dedekind sum]]). See [[#References|[a2]]], Chapt. IV, for a further discussion.
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R. Dedekind [[#References|[a1]]] comments on computations of B. Riemann in connection with [[theta-function]]s. He shows that it is basic to understand the transformation behaviour of the logarithm of the function now carrying his name. This study leads him to quantities now called [[Dedekind sum]]s. See [[#References|[a2]]], Chapt. IV, for a further discussion.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Dedekind,  "Erläuterungen zu den fragmenten XXVIII"  H. Weber (ed.) , ''B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass'' , Dover, reprint  (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Rademacher,  E. Grosswald,  "Dedekind sums" , Math. Assoc. America  (1972)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Dedekind,  "Erläuterungen zu den fragmenten XXVIII"  H. Weber (ed.) , ''B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass'' , Dover, reprint  (1953) {{ZBL|0053.19405}}</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Rademacher,  E. Grosswald,  "Dedekind sums" , Math. Assoc. America  (1972) {{ZBL|0251.10020}}</TD></TR>
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</table>
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[[Category:Special functions]]

Latest revision as of 19:54, 12 April 2017

2020 Mathematics Subject Classification: Primary: 11F20 [MSN][ZBL]

The function defined by

$$\eta(z)=e^{\pi iz/12}\prod_{n=1}^\infty(1-e^{2\pi inz})$$

for $z\in\mathbf C$, $\operatorname{Im}z>0$. As the infinite product converges absolutely, uniformly for $z$ in compact sets (cf. Uniform convergence), the function $\eta$ is holomorphic (cf. Analytic function). Moreover, it satisfies $\eta(z+1)=e^{\pi i/12}\eta(z)$ and $\eta(-1/z)=\sqrt{-iz}\eta(z)$. So, $\eta^{24}$ is a modular form of weight $12$ (cf. also Modular group).

R. Dedekind [a1] comments on computations of B. Riemann in connection with theta-functions. He shows that it is basic to understand the transformation behaviour of the logarithm of the function now carrying his name. This study leads him to quantities now called Dedekind sums. See [a2], Chapt. IV, for a further discussion.

References

[a1] R. Dedekind, "Erläuterungen zu den fragmenten XXVIII" H. Weber (ed.) , B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass , Dover, reprint (1953) Zbl 0053.19405
[a2] H. Rademacher, E. Grosswald, "Dedekind sums" , Math. Assoc. America (1972) Zbl 0251.10020
How to Cite This Entry:
Dedekind eta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_eta-function&oldid=32932
This article was adapted from an original article by R.W. Bruggeman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article