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Riemann theta functions with characteristics

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1: 21.2 Definitions
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§21.2(ii) Riemann Theta Functions with Characteristics
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21.2.5 θ ⁢ [ 𝜶 𝜷 ] ⁡ ( 𝐳 | 𝛀 ) = 𝐧 ℤ g e 2 ⁢ π ⁢ i ⁢ ( 1 2 ⁢ [ 𝐧 + 𝜶 ] 𝛀 [ 𝐧 + 𝜶 ] + [ 𝐧 + 𝜶 ] [ 𝐳 + 𝜷 ] ) .
►This function is referred to as a Riemann theta function with characteristics [ 𝜶 𝜷 ] . … ►
21.2.7 θ ⁢ [ 𝟎 𝟎 ] ⁡ ( 𝐳 | 𝛀 ) = θ ⁡ ( 𝐳 | 𝛀 ) .
►For given 𝛀 , there are 2 2 ⁢ g g -dimensional Riemann theta functions with half-period characteristics. …
2: 21.8 Abelian Functions
§21.8 Abelian Functions
►For every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
3: 21.3 Symmetry and Quasi-Periodicity
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§21.3(ii) Riemann Theta Functions with Characteristics
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21.3.4 θ ⁢ [ 𝜶 + 𝐦 1 𝜷 + 𝐦 2 ] ⁡ ( 𝐳 | 𝛀 ) = e 2 ⁢ π ⁢ i ⁢ 𝜶 𝐦 2 ⁢ θ ⁢ [ 𝜶 𝜷 ] ⁡ ( 𝐳 | 𝛀 ) .
► …For Riemann theta functions with half-period characteristics, ►
21.3.6 θ ⁢ [ 𝜶 𝜷 ] ⁡ ( 𝐳 | 𝛀 ) = ( 1 ) 4 ⁢ 𝜶 𝜷 ⁢ θ ⁢ [ 𝜶 𝜷 ] ⁡ ( 𝐳 | 𝛀 ) .
4: 21.1 Special Notation
►Uppercase boldface letters are g × g real or complex matrices. ►The main functions treated in this chapter are the Riemann theta functions θ ⁡ ( 𝐳 | 𝛀 ) , and the Riemann theta functions with characteristics θ ⁢ [ 𝜶 𝜷 ] ⁡ ( 𝐳 | 𝛀 ) . ►The function Θ ⁡ ( ϕ | 𝐁 ) = θ ⁡ ( ϕ / ( 2 ⁢ π ⁢ i ) | 𝐁 / ( 2 ⁢ π ⁢ i ) ) is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
5: 21.6 Products
►On using theta functions with characteristics, it becomes ►
21.6.4 j = 1 h θ ⁢ [ k = 1 h T j ⁢ k ⁢ 𝐜 k k = 1 h T j ⁢ k ⁢ 𝐝 k ] ⁡ ( k = 1 h T j ⁢ k ⁢ 𝐳 k | 𝛀 ) = 1 𝒟 g ⁢ 𝐀 𝒦 𝐁 𝒦 e 2 ⁢ π ⁢ i ⁢ j = 1 h 𝐛 j 𝐜 j ⁢ j = 1 h θ ⁢ [ 𝐚 j + 𝐜 j 𝐛 j + 𝐝 j ] ⁡ ( 𝐳 j | 𝛀 ) ,
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21.6.7 θ ⁢ [ 1 2 ⁢ [ 𝐜 1 + 𝐜 2 + 𝐜 3 + 𝐜 4 ] 1 2 ⁢ [ 𝐝 1 + 𝐝 2 + 𝐝 3 + 𝐝 4 ] ] ⁡ ( 𝐱 + 𝐲 + 𝐮 + 𝐯 2 | 𝛀 ) ⁢ θ ⁢ [ 1 2 ⁢ [ 𝐜 1 + 𝐜 2 𝐜 3 𝐜 4 ] 1 2 ⁢ [ 𝐝 1 + 𝐝 2 𝐝 3 𝐝 4 ] ] ⁡ ( 𝐱 + 𝐲 𝐮 𝐯 2 | 𝛀 ) ⁢ θ ⁢ [ 1 2 ⁢ [ 𝐜 1 𝐜 2 + 𝐜 3 𝐜 4 ] 1 2 ⁢ [ 𝐝 1 𝐝 2 + 𝐝 3 𝐝 4 ] ] ⁡ ( 𝐱 𝐲 + 𝐮 𝐯 2 | 𝛀 ) ⁢ θ ⁢ [ 1 2 ⁢ [ 𝐜 1 𝐜 2 𝐜 3 + 𝐜 4 ] 1 2 ⁢ [ 𝐝 1 𝐝 2 𝐝 3 + 𝐝 4 ] ] ⁡ ( 𝐱 𝐲 𝐮 + 𝐯 2 | 𝛀 ) = 1 2 g ⁢ 𝜶 1 2 ⁢ ℤ g / ℤ g 𝜷 1 2 ⁢ ℤ g / ℤ g e 2 ⁢ π ⁢ i ⁢ 𝜷 [ 𝐜 1 + 𝐜 2 + 𝐜 3 + 𝐜 4 ] ⁢ θ ⁢ [ 𝐜 1 + 𝜶 𝐝 1 + 𝜷 ] ⁡ ( 𝐱 | 𝛀 ) ⁢ θ ⁢ [ 𝐜 2 + 𝜶 𝐝 2 + 𝜷 ] ⁡ ( 𝐲 | 𝛀 ) ⁢ θ ⁢ [ 𝐜 3 + 𝜶 𝐝 3 + 𝜷 ] ⁡ ( 𝐮 | 𝛀 ) ⁢ θ ⁢ [ 𝐜 4 + 𝜶 𝐝 4 + 𝜷 ] ⁡ ( 𝐯 | 𝛀 ) .
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§21.6(ii) Addition Formulas
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21.6.8 θ ⁢ [ 𝜶 𝜸 ] ⁡ ( 𝐳 1 | 𝛀 ) ⁢ θ ⁢ [ 𝜷 𝜹 ] ⁡ ( 𝐳 2 | 𝛀 ) = 𝝂 ℤ g / ( 2 ⁢ ℤ g ) θ ⁢ [ 1 2 ⁢ [ 𝜶 + 𝜷 + 𝝂 ] 𝜸 + 𝜹 ] ⁡ ( 𝐳 1 + 𝐳 2 | 2 ⁢ 𝛀 ) ⁢ θ ⁢ [ 1 2 ⁢ [ 𝜶 𝜷 + 𝝂 ] 𝜸 𝜹 ] ⁡ ( 𝐳 1 𝐳 2 | 2 ⁢ 𝛀 ) .
6: 21.5 Modular Transformations
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§21.5(ii) Riemann Theta Functions with Characteristics
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21.5.9 θ ⁢ [ 𝐃 ⁢ 𝜶 𝐂 ⁢ 𝜷 + 1 2 ⁢ diag ⁡ [ 𝐂 ⁢ 𝐃 T ] 𝐁 ⁢ 𝜶 + 𝐀 ⁢ 𝜷 + 1 2 ⁢ diag ⁡ [ 𝐀 ⁢ 𝐁 T ] ] ⁡ ( [ [ 𝐂 ⁢ 𝛀 + 𝐃 ] 1 ] T ⁢ 𝐳 | [ 𝐀 ⁢ 𝛀 + 𝐁 ] ⁢ [ 𝐂 ⁢ 𝛀 + 𝐃 ] 1 ) = κ ⁡ ( 𝜶 , 𝜷 , 𝚪 ) ⁢ det [ 𝐂 ⁢ 𝛀 + 𝐃 ] ⁢ e π ⁢ i ⁢ 𝐳 [ [ 𝐂 ⁢ 𝛀 + 𝐃 ] 1 ⁢ 𝐂 ] 𝐳 ⁢ θ ⁢ [ 𝜶 𝜷 ] ⁡ ( 𝐳 | 𝛀 ) ,
7: 21.7 Riemann Surfaces
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§21.7(ii) Fay’s Trisecant Identity
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21.7.9 E ⁡ ( P 1 , P 2 ) = θ ⁢ [ 𝜶 𝜷 ] ⁡ ( P 1 P 2 𝝎 | 𝛀 ) / ( ζ ⁡ ( P 1 ) ⁢ ζ ⁡ ( P 2 ) ) ,
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§21.7(iii) Frobenius’ Identity
8: Bibliography
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  • C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson (1985) Chapter 16 of Ramanujan’s second notebook: Theta-functions and q -series. Mem. Amer. Math. Soc. 53 (315), pp. v+85.
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  • G. Allasia and R. Besenghi (1989) Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.
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  • G. E. Andrews (1966a) On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.
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  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
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  • T. M. Apostol (1985a) Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.