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1: 21.7 Riemann Surfaces
§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. …
§21.7(iii) Frobenius’ Identity
2: 25.1 Special Notation
k , m , n nonnegative integers.
The main function treated in this chapter is the Riemann zeta function ζ ( s ) . This notation was introduced in Riemann (1859). The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
3: 21.2 Definitions
§21.2(i) Riemann Theta Functions
For numerical purposes we use the scaled Riemann theta function θ ^ ( 𝐳 | 𝛀 ) , 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: 25.17 Physical Applications
§25.17 Physical Applications
Analogies exist between the distribution of the zeros of ζ ( s ) 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). …
5: 25.10 Zeros
§25.10(i) Distribution
The Riemann hypothesis states that all nontrivial zeros lie on this line. …
§25.10(ii) Riemann–Siegel Formula
Riemann also developed a technique for determining further terms. …
6: 25.18 Methods of Computation
§25.18(i) Function Values and Derivatives
The principal tools for computing ζ ( s ) are the expansion (25.2.9) for general values of s , and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for ζ ( 1 2 + i t ) . …Calculations relating to derivatives of ζ ( s ) and/or ζ ( s , a ) can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). …
§25.18(ii) Zeros
Most numerical calculations of the Riemann zeta function are concerned with locating zeros of ζ ( 1 2 + i t ) in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of ζ ( s ) lie on the critical line s = 1 2 . …
7: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

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

  • Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover ζ ( s ) for 0 s 1 (15D), ζ ( s + 1 ) for 0 s 1 (20D), and ln ξ ( 1 2 + i x ) 25.4) for 1 x 1 (20D). For errata see Piessens and Branders (1972).

  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals < x 2 and 2 x < , with s = 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 4 , 10 8 , 10 12 .

  • 8: 25.3 Graphics
    §25.3 Graphics
    See accompanying text
    Figure 25.3.1: Riemann zeta function ζ ( x ) and its derivative ζ ( x ) , 20 x 10 . Magnify
    See accompanying text
    Figure 25.3.2: Riemann zeta function ζ ( x ) and its derivative ζ ( x ) , 12 x 2 . Magnify
    See accompanying text
    Figure 25.3.4: Z ( t ) , 0 t 50 . Z ( t ) and ζ ( 1 2 + i t ) have the same zeros. … Magnify
    See accompanying text
    Figure 25.3.6: Z ( t ) , 10000 t 10050 . Magnify
    9: 25.16 Mathematical Applications
    §25.16 Mathematical Applications
    which is related to the Riemann zeta function by … The Riemann hypothesis is equivalent to the statement …
    §25.16(ii) Euler Sums
    which satisfies the reciprocity law …
    10: 25.6 Integer Arguments
    §25.6(i) Function Values
    25.6.3 ζ ( n ) = B n + 1 n + 1 , n = 1 , 2 , 3 , .
    25.6.4 ζ ( 2 n ) = 0 , n = 1 , 2 , 3 , .
    §25.6(ii) Derivative Values
    §25.6(iii) Recursion Formulas