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Dedekind sum

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1: 27.14 Unrestricted Partitions
27.14.10 A k ( n ) = h = 1 ( h , k ) = 1 k exp ( π i s ( h , k ) - 2 π i n h k ) ,
and s ( h , k ) is a Dedekind sum given by
27.14.11 s ( h , k ) = r = 1 k - 1 r k ( h r k - h r k - 1 2 ) .
Dedekind sums occur in the transformation theory of the Dedekind modular function η ( τ ) , defined by …where ε = exp ( π i ( ( ( a + d ) / ( 12 c ) ) - s ( d , c ) ) ) and s ( d , c ) is given by (27.14.11). …
2: 23.18 Modular Transformations
Here s ( d , c ) is a Dedekind sum. …
3: Bibliography
  • T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.
  • 4: 23.17 Elementary Properties
    23.17.6 η ( τ ) = n = - ( - 1 ) n q ( 6 n + 1 ) 2 / 12 .
    5: Bibliography R
  • H. Rosengren (1999) Another proof of the triple sum formula for Wigner 9 j -symbols. J. Math. Phys. 40 (12), pp. 6689–6691.
  • K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
  • R. Roy (2017) Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.