# Riemann

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##### 1: 21.7 Riemann Surfaces
###### §21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
In almost all applications, a Riemann theta function is associated with a compact Riemann surface. … is a Riemann matrix and it is used to define the corresponding Riemann theta function. …
##### 2: 25.1 Special Notation
 $k,m,n$ nonnegative integers. …
The main function treated in this chapter is the Riemann zeta function $\zeta\left(s\right)$. This notation was introduced in Riemann (1859). The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\operatorname{Li}_{2}\left(z\right)$, the polylogarithm $\operatorname{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 3: 21.2 Definitions
###### §21.2(i) Riemann Theta Functions
For numerical purposes we use the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. …
###### §21.2(ii) Riemann Theta Functions with Characteristics
It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor: …
##### 4: 21.10 Methods of Computation
###### §21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
• Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

• Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.

• Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

• ##### 5: 25.7 Integrals
###### §25.7 Integrals
For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).
##### 6: 25.17 Physical Applications
###### §25.17 Physical Applications
Analogies exist between the distribution of the zeros of $\zeta\left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …
##### 7: 21.3 Symmetry and Quasi-Periodicity
###### §21.3(ii) Riemann Theta Functions with Characteristics
…For Riemann theta functions with half-period characteristics, …
##### 8: 25.20 Approximations
• Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$, $5\leq s\leq 11$, $11\leq s\leq 25$, $25\leq s\leq 55$. Precision is varied, with a maximum of 20S.

• Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta\left(s+1\right)$ and $\zeta\left(s+k\right)$, $k=2,3,4,5,8$, for $0\leq s\leq 1$ (23D).

• Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta\left(s\right)$ for $0\leq s\leq 1$ (15D), $\zeta\left(s+1\right)$ for $0\leq s\leq 1$ (20D), and $\ln\xi\left(\tfrac{1}{2}+ix\right)$25.4) for $-1\leq x\leq 1$ (20D). For errata see Piessens and Branders (1972).

• Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty and $2\leq x<\infty$, with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$.

• ##### 9: 21.9 Integrable Equations
###### §21.9 Integrable Equations
Typical examples of such equations are the Korteweg–de Vries equation … Furthermore, the solutions of the KP equation solve the Schottky problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemann surface (Schottky (1903)). …
##### 10: 21.6 Products
###### §21.6(i) Riemann Identity
Then …This is the Riemann identity. On using theta functions with characteristics, it becomes …